(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 390578, 11653]*) (*NotebookOutlinePosition[ 392237, 11704]*) (* CellTagsIndexPosition[ 392193, 11700]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["An Introduction to Coding Theory"]], "Title", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ By Robin Sanders and Lawrence Stout, copyright 1998\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Initialization Cells."]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " When ", StyleBox["Mathematica", FontSlant->"Italic"], " askes you if you want to initialize the notebook, press the yes button." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " The initialization cells contain specialized functions that are used \ in this notebook. A complete list of the contents of the initialization \ cells is contained in the appendix."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Hidden Cells"]], "Section", Evaluatable->False, CellOpen->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Off[General::spell]\nNeeds[\"Graphics`Colors`\"]"]], "Input", CellLabel->"In[1]:=", InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Code for drawing cubes."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Clear[top,bottom, cross]"]], "Input", CellLabel->"In[3]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "top[{cenx_, ceny_, cenz_}, size_, color_] :=\n{Graphics3D[\n {color,\n \ Line[{{cenx - size/2,ceny - size/2,cenz + size/2},\n {cenx - \ size/2,ceny + size/2,cenz + size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx - size/2,ceny - size/2,cenz + size/2},\n \ {cenx + size/2,ceny - size/2,cenz + size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx + size/2,ceny + size/2,cenz + size/2},\n \ {cenx - size/2,ceny + size/2,cenz + size/2}}]\n }],\n\n \ Graphics3D[\n {color,\n Line[{{cenx + size/2,ceny + size/2,cenz + \ size/2},\n {cenx + size/2,ceny - size/2,cenz + size/2}}]\n \ }]\n}\n "]], "Input", CellLabel->"In[4]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "bottom[{cenx_, ceny_, cenz_}, size_, color_] :=\n{Graphics3D[\n {color,\ \n Line[{{cenx - size/2,ceny - size/2,cenz - size/2},\n \ {cenx - size/2,ceny + size/2,cenz - size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx - size/2,ceny - size/2,cenz - size/2},\n \ {cenx + size/2,ceny - size/2,cenz - size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx + size/2,ceny + size/2,cenz - size/2},\n \ {cenx - size/2,ceny + size/2,cenz - size/2}}]\n }],\n\n \ Graphics3D[\n {color,\n Line[{{cenx + size/2,ceny + size/2,cenz - \ size/2},\n {cenx + size/2,ceny - size/2,cenz - size/2}}]\n \ }]\n}\n "]], "Input", CellLabel->"In[6]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "cross[{cenx_, ceny_, cenz_}, size_, color_] :=\n{Graphics3D[\n {color,\n\ Line[{{cenx - size/2,ceny - size/2,cenz - size/2},\n {cenx \ - size/2,ceny - size/2,cenz + size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx - size/2,ceny + size/2,cenz - size/2},\n \ {cenx - size/2,ceny + size/2,cenz + size/2}}]\n }],\n Graphics3D[\n \ {color,\n Line[{{cenx + size/2,ceny + size/2,cenz - size/2},\n \ {cenx + size/2,ceny + size/2,cenz + size/2}}]\n }],\n\n \ Graphics3D[\n {color,\n Line[{{cenx + size/2,ceny - size/2,cenz - \ size/2},\n {cenx + size/2,ceny - size/2,cenz + size/2}}]\n \ }]\n}\n "]], "Input", CellLabel->"In[8]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[cube]\ncube[{cenx_, ceny_, cenz_}, size_, \n \ topcolor_, bottomcolor_, crosscolor_] :=\nShow[\n top[{cenx, ceny, cenz}, \ size, topcolor],\n bottom[{cenx, ceny, cenz}, size, bottomcolor],\n \ cross[{cenx, ceny, cenz}, size, crosscolor],\n DisplayFunction->Identity,\n\ Boxed->False]"]], "Input", CellLabel->"In[10]:=", InitializationCell->True, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "cube6D[c, b, s] makes a 6D cube with big dimension b and cubelet dimension \ s."]], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "Clear[cube6D]\ncube6D[c_, b_, s_] :=\nModule[{BigCube, Cubelet1, Cubelet2, \ Cubelet3, Cubelet4, Cubelet5,\n Cubelet6, Cubelet7, Cubelet8},\n\ BigCube = cube[{c,c,c}, b, Gold, Gold, Gold];\nCubelet1 = cube[{c - b/2, c - \ b/2, c - b/2}, s, Blue, Red, Green];\nCubelet2 = cube[{c - b/2, c + b/2, c - \ b/2}, s, Blue, Red, Green];\nCubelet3 = cube[{c - b/2, c - b/2, c + b/2}, s, \ Blue, Red, Green];\nCubelet4 = cube[{c + b/2, c - b/2, c - b/2}, s, Blue, \ Red, Green];\nCubelet5 = cube[{c + b/2, c + b/2, c - b/2}, s, Blue, Red, \ Green];\nCubelet6 = cube[{c + b/2, c - b/2, c + b/2}, s, Blue, Red, Green];\n\ Cubelet7 = cube[{c - b/2, c + b/2, c + b/2}, s, Blue, Red, Green];\nCubelet8 \ = cube[{c + b/2, c + b/2, c + b/2}, s, Blue, Red, Green];\nShow[BigCube, \ Cubelet1,\n Cubelet2, Cubelet3,\n Cubelet4, Cubelet5, Cubelet6, \ Cubelet7, Cubelet8,\n ViewPoint->{50.000, 30.000, 15.000},\n \ DisplayFunction->Identity]\n]\n "]], "Input", CellLabel->"In[12]:=", InitializationCell->True, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "cube4D[c, out, in,color] draws a tesseract with outside cube length out and \ inside cube length in"]], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "Clear[cube4D]\ncube4D[c_, out_, in_, color_] :=\nModule[{InsideCube, \ OutsideCube, crosslines},\nInsideCube = cube[{c,c,c}, in, color, color, \ color];\nOutsideCube = cube[{c,c,c}, out, color, color, color];\ncrosslines =\ \n{Graphics3D[\n {color,\n Line[{{c - in/2, c - in/2, c - in/2},\ \n {c - out/2, c - out/2, c - out/2}}]}],\n Graphics3D[\n \ {color,\n Line[{{c - in/2, c - in/2, c + in/2},\n {c - \ out/2, c - out/2, c + out/2}}]}], \nGraphics3D[\n {color,\n \ Line[{{c - in/2, c + in/2, c - in/2},\n {c - out/2, c + \ out/2, c - out/2}}]}], \nGraphics3D[\n {color,\n \ Line[{{c - in/2, c + in/2, c + in/2},\n {c - out/2, c + out/2, \ c + out/2}}]}], \nGraphics3D[\n {color,\n Line[{{c + in/2, c - \ in/2, c - in/2},\n {c + out/2, c - out/2, c - out/2}}]}],\n\ Graphics3D[\n {color,\n Line[{{c + in/2, c - in/2, c + in/2},\n \ {c + out/2, c - out/2, c + out/2}}]}],\nGraphics3D[\n {color,\ \n Line[{{c + in/2, c + in/2, c - in/2},\n {c + out/2, c \ + out/2, c - out/2}}]}],\nGraphics3D[\n {color,\n Line[{{c + in/2, \ c + in/2, c + in/2},\n {c + out/2, c + out/2, c + out/2}}]}]};\n\ Show[InsideCube, OutsideCube, crosslines,\n ViewPoint->{50.000, 30.000, \ 15.000},\n DisplayFunction->Identity]\n] \ "]], "Input", CellLabel->"In[15]:=", InitializationCell->True, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "cube7D[c, out, in, s, color, top, bot, cross] . . . draws a 7 D cube with \ center at {c, c, c}, outside bigcube length out, inside bigcube length in, \ and cubelet length s."]], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "cube7D[c_, out_, in_, s_, color_, topcolor_, botcolor_, crosscolor_] :=\n\ Module[{tesseract, Cubelet1, Cubelet2, Cubelet3, Cubelet4, Cubelet5,\n \ Cubelet6, Cubelet7, Cubelet8, Cubelet9, Cubelet10, Cubelet11,\n \ Cubelet12, Cubelet13, Cubelet14, Cubelet15, Cubelet16},\ntesseract = \ cube4D[c, out, in, color];\nCubelet1 = cube[{c - in/2, c - in/2, c - in/2}, \ s, topcolor, botcolor, crosscolor];\nCubelet2 = cube[{c - in/2, c + in/2, c - \ in/2}, s, topcolor, botcolor, crosscolor];\nCubelet3 = cube[{c - in/2, c - \ in/2, c + in/2}, s, topcolor, botcolor, crosscolor];\nCubelet4 = cube[{c + \ in/2, c - in/2, c - in/2}, s, topcolor, botcolor, crosscolor];\nCubelet5 = \ cube[{c + in/2, c + in/2, c - in/2}, s, topcolor, botcolor, crosscolor];\n\ Cubelet6 = cube[{c + in/2, c - in/2, c + in/2}, s, topcolor, botcolor, \ crosscolor];\nCubelet7 = cube[{c - in/2, c + in/2, c + in/2}, s, topcolor, \ botcolor, crosscolor];\nCubelet8 = cube[{c + in/2, c + in/2, c + in/2}, s, \ topcolor, botcolor, crosscolor];\nCubelet9 = cube[{c - out/2, c - out/2, c - \ out/2}, s, topcolor, botcolor, crosscolor];\nCubelet10 = cube[{c - out/2, c + \ out/2, c - out/2}, s, topcolor, botcolor, crosscolor];\nCubelet11 = cube[{c - \ out/2, c - out/2, c + out/2}, s, topcolor, botcolor, crosscolor];\nCubelet12 \ = cube[{c + out/2, c - out/2, c - out/2}, s, topcolor, botcolor, crosscolor];\ \nCubelet13 = cube[{c + out/2, c + out/2, c - out/2}, s, topcolor, botcolor, \ crosscolor];\nCubelet14 = cube[{c + out/2, c - out/2, c + out/2}, s, \ topcolor, botcolor, crosscolor];\nCubelet15 = cube[{c - out/2, c + out/2, c + \ out/2}, s, topcolor, botcolor, crosscolor];\nCubelet16 = cube[{c + out/2, c + \ out/2, c + out/2}, s, topcolor, botcolor, crosscolor];\nShow[tesseract, \ Cubelet1,\n Cubelet2, Cubelet3,\n Cubelet4, Cubelet5, Cubelet6, \ Cubelet7, Cubelet8,\n Cubelet9, Cubelet10, Cubelet11, Cubelet12, \ Cubelet13,\n Cubelet14, Cubelet15, Cubelet16,\n ViewPoint->{50.000, \ 30.000, 15.000},\n DisplayFunction->Identity]\n]\n "]], "Input", CellLabel->"In[17]:=", InitializationCell->True, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Additonal initialization cells for the Hamming 7-4 code."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Ham74[{x, y, z, w}] is the LT that does the encoding\nin the Hamming 7-4 \ code."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "Clear[Ham74,x,y,z]\nHam74[{x_, y_, z_, w_}] = {Mod[x + y + w, 2], \n \ Mod[x + z + w, 2], \n x, \n \ Mod[y + z + w, 2], \n y, z, \ w};"]], "Input", CellLabel->"In[19]:=", InitializationCell->True, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "infopoint4[{x1, x2, x3, x4}, c, out, in, color] makes a \"Point\" at the \ correct corner of the tesseract centered at {c,c,c} with outside dimension \ out, inside dimension in, and draws the point in the specified color."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "Clear[infopoint4]\ninfopoint4[{x1_, x2_, x3_, x4_}, c_, out_, in_, color_] \ :=\n Module[{center, outer, inner,corner, info},\n center = c;\n \ outer = out/2;\n inner = in/2;\n info = {x1, x2, x3, x4};\n\ If[info[[1]]==0, corner = outer, corner = inner];\n \ Graphics3D[{color, PointSize[.05],\n Point[{center + (-1)^(info[[4]] \ + 1) corner,\n center + (-1)^(info[[3]] + 1) corner,\n \ center + (-1)^(info[[2]] + 1) corner}]}]\n ]"]], "Input", CellLabel->"In[21]:=", InitializationCell->True, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "codepoint74[{x1, x2, x3, x4}, c, out, in, s, color] makes a \"Point\" at the \ correct corner of the 7D cube for the codeword corresponding to {x1,x2,x3,x4} \ centered at {c,c,c} with outside dimension out, inside dimension in, and \ draws the point in the specified color."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "codepoint74[{x1_, x2_, x3_, x4_}, c_, out_, in_, s_,color_] :=\n \ Module[{center, outer, inner,corner, s1, ham},\n center = c;\n \ outer = out/2;\n inner = in/2;\n s1 = s/2;\n ham = \ Ham74[{x1, x2, x3, x4}];\n If[ham[[1]]==0, corner = outer, corner = \ inner];\n Graphics3D[{color, PointSize[.02],\n Point[{center \ + (-1)^(ham[[4]] + 1) corner + (-1)^(ham[[7]] + 1) s1,\n \ center + (-1)^(ham[[3]] + 1) corner + (-1)^(ham[[6]] + 1) s1,\n \ center + (-1)^(ham[[2]] + 1) corner + (-1)^(ham[[5]] + 1) s1}]}]\n \ ]"]], "Input", CellLabel->"In[23]:=", InitializationCell->True, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Coloring neighbors of codewords---\nColorNeighbors74[{x1, x2, x3, x4}, c, \ out, in, s, color] colors all the neighbors of the codeword for \n{x1, x2, \ x3, x4} in the same color."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "Clear[ColorNeighbors]\nColorNeighbors74[{x_, y_, z_, w_}, c_, out_, in_, s_, \ color_] :=\nModule[{neigbors,a, k, points, codeword, ham, corner, s1, center, \ outer, inner},\n neighbors = { };\n codeword = Ham74[{x, y, z, w}];\n Do[{a = \ ReplacePart[codeword, \n Mod[codeword[[k]]+1, 2],k];\n \ AppendTo[neighbors, a];},\n {k,1, Length[codeword ] }];\n center = c;\n \ outer = out/2;\n inner = in/2;\n s1 = s/2;\n points = { };\n Do[{ham = \ neighbors[[k]];\n If[ham[[1]]==0, corner = outer, corner = inner];\n \ AppendTo[points,\n Graphics3D[{color, PointSize[.015],\n \ Point[{center + (-1)^(ham[[4]] + 1) corner + (-1)^(ham[[7]] + 1) s1,\n \ center + (-1)^(ham[[3]] + 1) corner + (-1)^(ham[[6]] + 1) s1,\n\ center + (-1)^(ham[[2]] + 1) corner + (-1)^(ham[[5]] + 1) \ s1}]}]]\n },\n {k, 1, Length[neighbors]}];\n Show[points, \ DisplayFunction->Identity]\n ]"]], "Input", CellLabel->"In[24]:=", InitializationCell->True, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Here are the standard \"pictures\" of the\n3-dimensional, 4-dimensional and \ 7-dimensional cubes."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]], Cell[TextData[StyleBox[ "ThreeCube = cube[{1,1,1},2,Blue,Red, Green];"]], "Input", CellLabel->"In[26]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "labels = {\n Graphics3D[Text[\" \",{1,-.25,1}]],\n \ Graphics3D[Text[\"{0,0,0}\", {0,.45,.25}]],\n \ Graphics3D[Text[\"{0,0,1}\", {2,.15,-.15}]],\n \ Graphics3D[Text[\"{0,1,0}\", {0,2,-.25}]],\n Graphics3D[Text[\"{1,0,0}\", \ {0,0,2.15}]],\n Graphics3D[Text[\"{0,1,1}\", {2,2,-.15}]],\n \ Graphics3D[Text[\"{1,1,0}\", {0,2,2.15}]],\n Graphics3D[Text[\"{1,0,1}\", \ {2,0.15,1.85}]],\n Graphics3D[Text[\"{1,1,1}\", {2,2.45,2}]]};"]], "Input",\ CellLabel->"In[27]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "LabeledThreeCube =\nShow[ThreeCube,labels,\nViewPoint->{2.939, 1.262, \ 1.104},\n DisplayFunction->Identity];"]], "Input", CellLabel->"In[28]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "c = 10;\nout = 20;\nin = 7;\ns = 1.5;\nFourCube = cube4D[c, out, in,Bisque];\ \nSevenCube=cube7D[c,out,in,s,Bisque,Blue,Red,Green];"]], "Input", CellLabel->"In[29]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "MonoSevenCube=cube7D[c,out,in,s,Gray,Gray,\n \ Gray,Gray ];\n"]], "Input", CellLabel->"In[35]:=", InitializationCell->True, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The information words and the code words---drawing stuff."]], "Subsubsection",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info0 = infopoint4[{0,0,0,0}, c, out, in, Red];\ncode0 = \ codepoint74[{0,0,0,0}, c, out, in, s, Red];\nneigh0 = \ ColorNeighbors74[{0,0,0,0}, c, out, in, s, Red];"]], "Input", CellLabel->"In[36]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info1 = infopoint4[{0,0,0,1},c, out, in,Blue];\ncode1 = \ codepoint74[{0,0,0,1},c, out, in,s,Blue];\nneigh1 = \ ColorNeighbors74[{0,0,0,1},c, out, in,s,Blue];"]], "Input", CellLabel->"In[39]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info2 = infopoint4[{0,0,1,0},c, out, in,Green];\ncode2 = \ codepoint74[{0,0,1,0},c, out, in,s,Green];\nneigh2 = \ ColorNeighbors74[{0,0,1,0},c, out, in,s,Green];"]], "Input", CellLabel->"In[42]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info3 = infopoint4[{0,0,1,1},c, out, in,Yellow];\ncode3 = \ codepoint74[{0,0,1,1},c, out, in,s,Yellow];\nneigh3 = \ ColorNeighbors74[{0,0,1,1},c, out, in,s,Yellow];"]], "Input", CellLabel->"In[45]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info4 = infopoint4[{0,1,0,0},c, out, in,Pink];\ncode4 = \ codepoint74[{0,1,0,0},c, out, in,s,Pink];\nneigh4 = \ ColorNeighbors74[{0,1,0,0},c, out, in,s,Pink];"]], "Input", CellLabel->"In[48]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info5 = infopoint4[{0,1,0,1},c, out, in,Brick];\ncode5 = \ codepoint74[{0,1,0,1},c, out, in,s,Brick];\nneigh5 = \ ColorNeighbors74[{0,1,0,1},c, out, in,s,Brick];"]], "Input", CellLabel->"In[51]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info6 = infopoint4[{0,1,1,0},c, out, in,Cadet];\ncode6 = \ codepoint74[{0,1,1,0},c, out, in,s,Cadet];\nneigh6 = \ ColorNeighbors74[{0,1,1,0},c, out, in,s,Cadet];"]], "Input", CellLabel->"In[54]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info7 = infopoint4[{0,1,1,1},c, out, in,Orange];\ncode7 = \ codepoint74[{0,1,1,1},c, out, in,s,Orange];\nneigh7 = \ ColorNeighbors74[{0,1,1,1},c, out, in,s,Orange];"]], "Input", CellLabel->"In[57]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info8 = infopoint4[{1,0,0,0}, c, out, in,ChromeOxideGreen];\ncode8 = \ codepoint74[{1,0,0,0}, c, out, in,s,ChromeOxideGreen];\nneigh8 = \ ColorNeighbors74[{1,0,0,0}, c, out, in,s,ChromeOxideGreen];"]], "Input", CellLabel->"In[60]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info9 = infopoint4[{1,0,0,1}, c, out, in,CobaltVioletDeep];\ncode9 = \ codepoint74[{1,0,0,1}, c, out, in,s,CobaltVioletDeep];\nneigh9 = \ ColorNeighbors74[{1,0,0,1}, c, out, in,s,CobaltVioletDeep];"]], "Input", CellLabel->"In[63]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info10 = infopoint4[{1,0,1,0}, c, out, in,Cyan];\ncode10 = \ codepoint74[{1,0,1,0}, c, out, in,s,Cyan];\nneigh10 = \ ColorNeighbors74[{1,0,1,0}, c, out, in,s,Cyan];"]], "Input", CellLabel->"In[66]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info11 = infopoint4[{1,0,1,1}, c, out, in,EnglishRed];\ncode11 = \ codepoint74[{1,0,1,1}, c, out, in,s,EnglishRed];\nneigh11 = \ ColorNeighbors74[{1,0,1,1}, c, out, in,s,EnglishRed];"]], "Input", CellLabel->"In[69]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "info12 = infopoint4[{1,1,0,0},c, out, in, Goldenrod];\ncode12 = \ codepoint74[{1,1,0,0}, c, out, in,s,Goldenrod];\nneigh12 = \ ColorNeighbors74[{1,1,0,0}, c, out, in,s,Goldenrod];"]], "Input", CellLabel->"In[72]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ 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"Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[TextToBinary]\nTextToBinary[text_,bits_] := \n\ Module[{characters,step2},\n characters = ToCharacterCode[text];\n \ step2 = IntegerDigits[characters, 2];\n Table[\n \ Take[Flatten[Prepend[step2[[k]],{0,0,0}]],\n \ -bits],\n {k, 1,Length[step2]}]\n ] "]], "Input", CellLabel->"In[84]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[InfoBits]\nInfoBits[bitstream_,infobits_] := \nModule[{total, zeros, \ remainder,k,\n fatbits},\n\ttotal = Length[bitstream];\n If[ \ Mod[total, infobits] != 0, \n {remainder = Mod[total, infobits];\n\ zeros = Flatten[\n Table[0,{k, infobits - \ remainder}]];\n fatbits = Flatten[Append[bitstream, zeros]];\n \ },\n fatbits = bitstream;\n ];\n \ Partition[fatbits, infobits]\n ]"]], "Input", CellLabel->"In[86]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[AddErrors]\nAddErrors[bitstream_, num_] := \n\tModule[{errorbits, \ receivedbitstream},\n\t receivedbitstream = bitstream;\n\t errorbits = \ Table[Random[Integer, {1, Length[bitstream]}],\n\t \ {k,1,num}];\n Do[\n receivedbitstream = \ ReplacePart[receivedbitstream, \n \ Mod[receivedbitstream[[errorbits[[k]]]]+1, 2],\n errorbits[[k]]\ \n ],\n {k,1, Length[errorbits]}];\n \ receivedbitstream\n ]"]], "Input", CellLabel->"In[88]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[BitstreamToBinary]\nBitstreamToBinary[bitstream_, bits_] := \n\ Module[{extra, bitstream1},\n\textra = Mod[Length[bitstream], bits];\n \ bitstream1 = Drop[bitstream, -extra];\n Partition[bitstream1, bits] \n \ ]"]], "Input", CellLabel->"In[90]:=", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[BinaryToText]\nBinaryToText[binary_, bits_] := \n\ Module[{twopowers,characters, k},\n twopowers = Table[2^{bits-k}, {k, 1, \ bits}];\n characters =\n Flatten[\n \ Table[binary[[k]].twopowers, {k, 1, Length[binary]}]\n ];\n \ FromCharacterCode[characters]\n ]"]], "Input", CellLabel->"In[92]:=", InitializationCell->True, AspectRatioFixed->True] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Basic outline of problem"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " In a computer, each character of ASCII text is stored as a 7 or 8 bit \ \"string\" of 0's and 1's. The entire text string itself is thus stored as a \ very long string of 0's and 1's that is called a \"bitstream.\" For \ example, the following cell converts the characters in the text between the \ quote marks to 7-bit binary words. "]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[binary];\nbinary = TextToBinary[\"A sample!\", 7]"]], "Input", CellLabel->"In[94]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 1, 1}, { 1, 1, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0}, { 1, 1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 1, 0, 1}, {0, 1, 0, 0, 0, 0, 1}} \)], "Output", CellLabel->"Out[95]="] }, Open ]], Cell[TextData[{ StyleBox[ "Each of these words can be thought of as a vector over the field "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\)]]], StyleBox[ ". The bitstream that is used to store the text is formed from the list of \ binary words by deleting the braces that are used separate the words. Here's \ the bitstream for the text:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[bitstream];\nbitstream = Flatten[binary]"]], "Input", CellLabel->"In[96]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1}\)], "Output", CellLabel->"Out[97]="] }, Open ]], Cell[TextData[StyleBox[ " When transmitting a bitstring across communication lines, errors in the \ bitstring can occur. In other words, some of the 0's may be transmitted as \ 1's and vice versa. The function, AddErrors[ ], mimics this situation. For \ example, if one random error occurs in the transmission of the bitstream, the \ recieved bitstream looks like:"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[receivedbitstream];\nreceivedbitstream = AddErrors[bitstream, 1]"]], "Input", CellLabel->"In[98]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1}\)], "Output", CellLabel->"Out[99]="] }, Open ]], Cell[TextData[StyleBox[ "If one or more errors occurs during transmission, the bitstring cannot be \ correctly reconverted to text at the receiving end, as is shown in the output \ of the following cell. "]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[receivedbinary, receivedtext]\nreceivedbinary = \n \ BitstreamToBinary[receivedbitstream, 7]\nreceivedtext = \n \ BinaryToText[receivedbinary, 7]"]], "Input", CellLabel->"In[100]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 1, 1}, { 1, 1, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0}, { 1, 1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 1, 0, 1}, {0, 1, 0, 0, 0, 0, 1}} \)], "Output", CellLabel->"Out[101]="], Cell[BoxData[ \("A sa}ple!"\)], "Output", CellLabel->"Out[102]="] }, Open ]], Cell[TextData[StyleBox[ " Our task: Use linear algebra and some geometry to design a way to \ transmit the bitstream so that we can detect (and correct) errors that occur \ during transmission."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " This problem was first considered by Richard Hamming in 1948 when he \ grew frustrated with the fact that the vaccumn tube computer he was working \ with would simply stop working on his programs whenever a transmission error \ occured. We'll look at his solution in detail in this notebook."]], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Hamming's idea"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " First, Hamming chopped the bitstream up into 4-bit \"information \ words.\" Each of these words is an element of "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[ " and can be thought of as a vertex in a 4 dimensional cube (or a \ tesseract) where adjacent vertices differ in exactly one coordinate. \ Clearly, an error in any bit transforms one information word into another \ information word\[LongDash]that's the problem we want to get around. \ Hamming's idea was to add extra \"check bits\" to each word in such a way as \ to both detect occasonal errors and to correct those errors if possible."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " Mathematically, the process of adding the extra bits to the code words \ is the same as finding a map from the vector space "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[" to a larger dimensional vector space over the field "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\)]]], StyleBox[ ". Linear transformations are particularly nice mappings, so Hamming's \ basic idea can be thought of in the following way: We need to find a 1-1 \ linear transformation from "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[ " to a larger dimensional vector space with a special property. If we call \ the set of vectors in the image of the linear transformation \"codewords,\" \ then we want the linear transformation to increase the \"distance\" between \ the codewords so that an error in one bit does not transform one codeword \ (the image of an information word) into another one. If we can do this, then \ we should be able to detect when errors in transmission occur. If we pick \ the codomain and the linear transformation carefully, we may even be able to \ correct the errors we detect."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " One of the nicest codes that Hamming discovered uses a linear \ transformation from "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[" to "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^7\)]]], StyleBox[ " and is called the Hamming(7,4) code. We will study this particular \ example in depth."] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The Geometry of three, four, and seven dimensional cubes"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " In order to understand the correct idea of \"distance\" in the problem, \ we need some definitions. After the definitions, we need to look at some \ geometry. "]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Definition. ", FontWeight->"Bold"], StyleBox["The"], StyleBox[" n-dimensional binary cube", FontWeight->"Bold"], StyleBox[" is the set of vectors in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^n\)]]], StyleBox["."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Definition. 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0.00639264, 0.00639264}}] }, Open ]], Cell[TextData[StyleBox[ " The linear transformation that encodes the 4 bit information \"words\" \ as 7-bit \"codewords\" is given by"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Ham74[{x, y, z, w}]"]], "Input", CellLabel->"In[112]:=", AspectRatioFixed->True], Cell[BoxData[ \({Mod[w + x + y, 2], Mod[w + x + z, 2], x, Mod[w + y + z, 2], y, z, w} \)], "Output", CellLabel->"Out[112]="] }, Open ]], Cell[TextData[{ StyleBox["Mod[*, 2] is "], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox["'s way of represnting the arithmetic in the field "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\)]]], StyleBox[ ". So the check bits (bits 1, 2, and 4) are formed from information \ bits (bits 3, 5, 6, and 7) by simply adding certain triples of information \ bits together in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\)]]], StyleBox["."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" The nice thing about "], StyleBox[Cell[BoxData[ \(TraditionalForm\`Ham74[{x, y, z, w}]\)]]], StyleBox[ " is that after we do the coding, the distance between the colored \ vertices is increased. 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That tells us that we can correct single bit errors \ by taking non-codewords to the nearest codeword. The picture also shows that \ every point in the 7-cube has been colored. This tells us that every point \ is within Hamming distance 1 of a codeword. Such codes are called "], StyleBox["perfect.", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Linear Algebra for Error Correction"]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " We can determine whether an error occured by noticing that the \ following linear transformation, called ", StyleBox["syndrome", FontWeight->"Bold"], ", sends all the codewords (and only the codewords) to the zero vector. \ This makes the space of codewords the kernel of the linear transformation \ syndrome." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[syndrome]\nsyndrome[\n{check1_,check2_,x_,check3_,y_,z_,w_}]=\n\t \ {Mod[check1 + w + x + y, 2], \n Mod[check2 + w + x + z, 2],\n \ Mod[check3 + w + y + z, 2]}"]], "Input", CellLabel->"In[116]:=", AspectRatioFixed->True], Cell[BoxData[ \({Mod[check1 + w + x + y, 2], Mod[check2 + w + x + z, 2], Mod[check3 + w + y + z, 2]}\)], "Output", CellLabel->"Out[117]="] }, Open ]], Cell[TextData[StyleBox[ "The placement of the checksums in the Hamming(7,4) code is a clever one. \ Errors can happen in any bit, so looking at what the syndrome gives for one \ bit errors may be useful. We'll look at vectors adjacent to zero."]], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "MatrixForm[{syndrome[{1,0,0,0,0,0,0}], \n \ syndrome[{0,1,0,0,0,0,0}],\n\t\t\tsyndrome[{0,0,1,0,0,0,0}], \n\t\t\t\ syndrome[{0,0,0,1,0,0,0}],\n\t\t\tsyndrome[{0,0,0,0,1,0,0}], \n\t\t\t\ syndrome[{0,0,0,0,0,1,0}],\n\t\t\tsyndrome[{0,0,0,0,0,0,1}]}]"]], "Input", CellLabel->"In[118]:=", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"0", "1", "0"}, {"1", "1", "0"}, {"0", "0", "1"}, {"1", "0", "1"}, {"0", "1", "1"}, {"1", "1", "1"} }], ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[118]//MatrixForm="] }, Open ]], Cell[TextData[{ StyleBox["Notice that "], StyleBox["syndrome", FontSlant->"Italic"], StyleBox["["], StyleBox["v", FontWeight->"Bold"], StyleBox[ "] give the binary representation of the place where the 1 was, when we \ read from low bit to high bit. Thus we can use the syndrome to both \ determine if an error occured ("], StyleBox["syndrome", FontSlant->"Italic"], StyleBox["["], StyleBox["v", FontWeight->"Bold"], StyleBox[ "] is not equal to the zero vector) and where the error occured. In other \ words, "], StyleBox["syndrome", FontSlant->"Italic"], StyleBox["["], StyleBox["v", FontWeight->"Bold"], StyleBox[ "] tells us which bit to change when an error is detected. Coding the \ error correction scheme into "], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" takes several steps."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" First, we use "], StyleBox["syndrome", FontSlant->"Italic"], StyleBox[" to detect whether an error has occured. The function "], StyleBox["noerrordetect", FontSlant->"Italic"], StyleBox[" returns True if no error is detected (i.e. "], StyleBox["v", FontWeight->"Bold"], StyleBox[ " is a codeword) and False if a single error is detected (i.e. "], StyleBox["v", FontWeight->"Bold"], StyleBox[" is not a codeword):"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Clear[noerrordetect]\nnoerrordetect[\n {x1_,x2_,x3_,x4_,x5_,x6_,x7_}]:= \n\ syndrome[{x1,x2,x3,x4,x5,x6,x7}]=={0,0,0}"]], "Input", CellLabel->"In[119]:=", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "When an error is detected, we determine which bit to change by converting \ "], StyleBox["syndrome", FontSlant->"Italic"], StyleBox["["], StyleBox["v", FontWeight->"Bold"], StyleBox["] to a number:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "bittochange[{x1_,x2_,x3_,x4_,x5_,x6_,x7_}]:=\n \t\ syndrome[{x1,x2,x3,x4,x5,x6,x7}][[1]] +\n \t2 \ syndrome[{x1,x2,x3,x4,x5,x6,x7}][[2]] +\n \t4 \ syndrome[{x1,x2,x3,x4,x5,x6,x7}][[3]] "]], "Input", CellLabel->"In[121]:=", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Finally, we change the bit with the error. Given a positive integer n and \ a vector "], StyleBox["v", FontWeight->"Bold"], StyleBox[", the function "], StyleBox["changebit", FontSlant->"Italic"], StyleBox["["], StyleBox["n", FontSlant->"Italic"], StyleBox[", "], StyleBox["v", FontWeight->"Bold"], StyleBox["] changes the "], StyleBox["n", FontSlant->"Italic"], StyleBox["th bit of the vector from 0 to 1 or vice versa:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "changebit[n_,\n {x1_,x2_,x3_,x4_,x5_,x6_,x7_}]:=\nReplacePart[{x1, \ x2, x3, x4, x5, x6, x7},\n Mod[1 + \n {x1, x2, x3, x4, x5, \ x6, x7}[[n]],\n 2],\n \t\t n]"]], "Input", CellLabel->"In[122]:=", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We now put the pieces of the error-detection/error-correction scheme \ together into one "], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" function, "], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ StyleBox["errorcorrect", FontSlant->"Italic"], "[", StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "]"}], TraditionalForm]]]], StyleBox[". Note "], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ StyleBox["errorcorrect", FontSlant->"Italic"], "[", StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "]"}], TraditionalForm]]]], StyleBox[" does nothing if "], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ StyleBox["syndrome", FontSlant->"Italic"], "[", StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "]"}], TraditionalForm]]]], StyleBox[" equals the zero vector (i.e. "], StyleBox["v", FontWeight->"Bold"], StyleBox[ " is a codeword) and corrects the detected error by changing the bit \ indicated by "], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ StyleBox["syndrome", FontSlant->"Italic"], "[", StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "]"}], TraditionalForm]]]], StyleBox[" when an error is detected:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "errorcorrect[\n{x1_,x2_,x3_,x4_,x5_,x6_,x7_}]:=\n \ If[noerrordetect[{x1,x2,x3,x4,x5,x6,x7}],\n {x1,x2,x3,x4,x5,x6,x7}, \n \ changebit[\n bittochange[{x1,x2,x3,x4,x5,x6,x7}],\n \ {x1,x2,x3,x4,x5,x6,x7}]]"]], "Input", CellLabel->"In[123]:=", AspectRatioFixed->True], Cell[TextData[{ StyleBox[" Here's a demonstration of how"], StyleBox[" errorcorrect", FontSlant->"Italic"], StyleBox[ " works: The vector {0, 0, 1, 1, 0, 0, 1} is a code word since it equals \ "], StyleBox[Cell[BoxData[ \(TraditionalForm\`Ham74[{1, 0, 0, 1}]\)]]], StyleBox[":"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Ham74[{1,0,0,1}]"]], "Input", CellLabel->"In[124]:=", AspectRatioFixed->True], Cell[BoxData[ \({0, 0, 1, 1, 0, 0, 1}\)], "Output", CellLabel->"Out[124]="] }, Open ]], Cell[TextData[{ StyleBox["The function "], StyleBox["errorcorrect", FontSlant->"Italic"], StyleBox[ " leaves {0, 0, 1, 1, 0, 0, 1} alone because this vector is a codeword:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["errorcorrect[{0, 0, 1, 1, 0, 0, 1}]"]], "Input", CellLabel->"In[125]:=", AspectRatioFixed->True], Cell[BoxData[ \({0, 0, 1, 1, 0, 0, 1}\)], "Output", CellLabel->"Out[125]="] }, Open ]], Cell[TextData[{ StyleBox["But "], StyleBox["errorcorrect", FontSlant->"Italic"], StyleBox[ " takes each of the neighbors of the codeword {0, 0, 1, 1, 0, 0, 1} to this \ codeword:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "MatrixForm[\n{errorcorrect[{1, 0, 1, 1, 0, 0, 1}],\n errorcorrect[{0, 1, 1, \ 1, 0, 0, 1}],\n errorcorrect[{0, 0, 0, 1, 0, 0, 1}],\n errorcorrect[{0, 0, 1, \ 0, 0, 0, 1}],\n errorcorrect[{0, 0, 1, 1, 1, 0, 1}],\n errorcorrect[{0, 0, 1, \ 1, 0, 1, 1}],\n errorcorrect[{0, 0, 1, 1, 0, 0, 0}]}]"]], "Input", CellLabel->"In[126]:=", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"}, {"0", "0", "1", "1", "0", "0", "1"} }], ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[126]//MatrixForm="] }, Open ]], Cell[TextData[{ StyleBox["In general, "], StyleBox["errorcorrect", FontSlant->"Italic"], StyleBox[ " takes any vertex of the seven cube back to the codeword to which it is \ closest. Since each vertex of the seven cube is either a code word or \ adjacent to a codeword, "], StyleBox["errorcorrect", FontSlant->"Italic"], StyleBox[ " can both detect and correct a one bit error in each transmitted code \ word."] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Text To Binary To Bitstream & Back--an entire example"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " We now go back to the original problem: How to tranmit a string of \ text and correct at least some of the errors."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Convert text to bitstream."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " The first step is to convert the text string to 7 bit binary words and \ then \"flatten\" the sequence of 7-coordinate vectors into a bitstream."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Step1: Convert the text into a string of binary vectors. "]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[binary];\nbinary = TextToBinary[\"A sample!\", 7]"]], "Input", CellLabel->"In[127]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 1, 1}, { 1, 1, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0}, { 1, 1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 1, 0, 1}, {0, 1, 0, 0, 0, 0, 1}} \)], "Output", CellLabel->"Out[128]="] }, Open ]], Cell[TextData[StyleBox[ "Step 2. Convert binary vectors to bitstream using the Flatten command."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[bitstream];\nbitstream = Flatten[binary]"]], "Input", CellLabel->"In[129]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1}\)], "Output", CellLabel->"Out[130]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Code the bitstream."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " We now use the Hamming(7, 4) code to alter the bitstream we plan on \ transmitting. This involves several steps."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Step 1. Preparing to code"]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "We must split the bitstream up into \"information words\" of the correct \ size."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[infobitwords];\ninfobitwords = InfoBits[bitstream, 4]"]], "Input", CellLabel->"In[131]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0, 0, 1, 1}, {1, 0, 0, 1}, { 1, 1, 1, 0}, {0, 0, 0, 1}, {1, 1, 0, 1}, {1, 0, 1, 1}, {1, 1, 0, 0}, { 0, 0, 1, 1}, {0, 1, 1, 0}, {0, 1, 1, 0}, {0, 1, 0, 1}, {0, 1, 0, 0}, { 0, 0, 1, 0}}\)], "Output", CellLabel->"Out[132]="] }, Open ]], Cell[TextData[StyleBox["Step 2. Coding"]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We use the linear transformation "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox[" to get the spreadout codewords that live in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^7\)]]], StyleBox["."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[codewords];\ncodewords = Map[Ham74, infobitwords]"]], "Input", CellLabel->"In[133]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0}, {1, 1, 1, 0, 0, 0, 0}, { 1, 0, 0, 0, 0, 1, 1}, {0, 0, 1, 1, 0, 0, 1}, {0, 0, 1, 0, 1, 1, 0}, { 1, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1}, {0, 1, 1, 0, 0, 1, 1}, { 0, 1, 1, 1, 1, 0, 0}, {1, 0, 0, 0, 0, 1, 1}, {1, 1, 0, 0, 1, 1, 0}, { 1, 1, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 1}, {1, 0, 0, 1, 1, 0, 0}, { 0, 1, 0, 1, 0, 1, 0}}\)], "Output", CellLabel->"Out[134]="] }, Open ]], Cell[TextData[StyleBox["Step 3. Preparing to transmit"]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "We use Flatten to convert the codewords to coded bitstream, which will then \ be transmitted."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[codedbitstream];\ncodedbitstream = Flatten[codewords]"]], "Input", CellLabel->"In[135]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0}\)], "Output", CellLabel->"Out[136]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Add error and see what is received ."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " We use the AddErrors[ ] function to mimic the creation of random \ errors that occur during transmission. If one random error occurs in the \ transmission of the bitstream, the recieved bitstream looks like:"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[receivedbitstream];\nreceivedbitstream = \n \ AddErrors[codedbitstream, 1]"]], "Input", CellLabel->"In[137]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0}\)], "Output", CellLabel->"Out[138]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Detect and correct the errors."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " To detect and correct transmission errors we want to break the \ received bitstream up into received codewords\[LongDash]each of which is a \ 7-coordinate binary vector. Then we need to run each received codeword \ through the function "], StyleBox["errorcorrect", FontSlant->"Italic"], StyleBox[" that we defined earlier. "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Step 1. Partition the recieved bitstream into received codewords."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[receivedcodewords];\nreceivedcodewords = \n \ Partition[receivedbitstream, 7]"]], "Input", CellLabel->"In[139]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0}, {1, 1, 1, 0, 0, 0, 0}, { 1, 0, 0, 0, 0, 1, 1}, {0, 0, 1, 1, 0, 0, 1}, {0, 0, 1, 0, 1, 1, 0}, { 1, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1}, {0, 1, 1, 0, 0, 1, 1}, { 0, 1, 1, 1, 1, 0, 0}, {1, 0, 0, 0, 0, 1, 1}, {1, 1, 0, 0, 1, 1, 0}, { 1, 1, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 1}, {1, 0, 0, 1, 1, 0, 1}, { 0, 1, 0, 1, 0, 1, 0}}\)], "Output", CellLabel->"Out[140]="] }, Open ]], Cell[TextData[StyleBox["Step 2. Error correction. "]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" Run each received code word through the function "], StyleBox["errordetect", FontSlant->"Italic"], StyleBox[ " to correct any single bit errors that have occured. We'll do this to \ the whole list of received code words by using the Map command and call the \ result the list of corrected code words."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "CorrectedCodeWords = \n Map[errorcorrect, receivedcodewords]"]], "Input", CellLabel->"In[141]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0}, {1, 1, 1, 0, 0, 0, 0}, { 1, 0, 0, 0, 0, 1, 1}, {0, 0, 1, 1, 0, 0, 1}, {0, 0, 1, 0, 1, 1, 0}, { 1, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1}, {0, 1, 1, 0, 0, 1, 1}, { 0, 1, 1, 1, 1, 0, 0}, {1, 0, 0, 0, 0, 1, 1}, {1, 1, 0, 0, 1, 1, 0}, { 1, 1, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 1}, {1, 0, 0, 1, 1, 0, 0}, { 0, 1, 0, 1, 0, 1, 0}}\)], "Output", CellLabel->"Out[141]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Convert the corrected code words back to bitstream"]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ " Two steps are required to convert the list of corrected code words back \ to a bitstream. First, we must \"decode\" the corrected code words to \ recover a sequence of information 4-bit words. Then we must \"Flatten\" the \ sequence of information bit words to get the bitstream."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Step 1) Decode the corrected code words"]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " If no more than one transmission error was made in the transmission \ of each code word, then the vectors in CorrectedCodeWords and the vectors in \ codewords should be equal to each other. Since "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox[" is a one-to-one linear transformation from "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[" to "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^7\)]]], StyleBox[", there is a left inverse linear transformation that takes "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^7\)]]], StyleBox[" to "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[". We'll call this inverse transformation "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox[". (Saying that "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox[" is a left inverse of "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox[" means that "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox["[ "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox["[{"], StyleBox["x", FontSlant->"Italic"], StyleBox[", "], StyleBox["y", FontSlant->"Italic"], StyleBox[", "], StyleBox["z", FontSlant->"Italic"], StyleBox[", "], StyleBox["w", FontSlant->"Italic"], StyleBox["}] ] = {"], StyleBox["x", FontSlant->"Italic"], StyleBox[", "], StyleBox["y", FontSlant->"Italic"], StyleBox[", "], StyleBox["z", FontSlant->"Italic"], StyleBox[", "], StyleBox["w", FontSlant->"Italic"], StyleBox["} "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["for all the vectors in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox["."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" Recall that "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox["[{"], StyleBox["x", FontSlant->"Italic"], StyleBox[", "], StyleBox["y", FontSlant->"Italic"], StyleBox[", "], StyleBox["z", FontSlant->"Italic"], StyleBox[", "], StyleBox["w", FontSlant->"Italic"], StyleBox["}] is given by:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Ham74[{x, y, z, w}]"]], "Input", CellLabel->"In[142]:=", AspectRatioFixed->True], Cell[BoxData[ \({Mod[w + x + y, 2], Mod[w + x + z, 2], x, Mod[w + y + z, 2], y, z, w} \)], "Output", CellLabel->"Out[142]="] }, Open ]], Cell[TextData[{ StyleBox["To convert the codewords\[LongDash]that live in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^7\)]]], StyleBox[ "\[LongDash]back to the corresponding information bit words\[LongDash]that \ live in "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_2\^4\)]]], StyleBox[ "\[LongDash]all we have to do is \"throw away\" the checkbit coordinates. \ In other words, "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox[" is given by:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[DecodeH74]\nDecodeH74[\n{check1_,check2_,x_,check3_,y_,z_,w_}]=\n \ {x, y, z, w}"]], "Input", CellLabel->"In[143]:=", AspectRatioFixed->True], Cell[BoxData[ \({x, y, z, w}\)], "Output", CellLabel->"Out[144]="] }, Open ]], Cell[TextData[{ StyleBox["We can check that "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox[" really is the left inverse of "], StyleBox["Ham74", FontSlant->"Italic"], StyleBox[" by doing the appropriate composition:"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["DecodeH74[Ham74[{x, y, z, w}]]"]], "Input", CellLabel->"In[145]:=", AspectRatioFixed->True], Cell[BoxData[ \({x, y, z, w}\)], "Output", CellLabel->"Out[145]="] }, Open ]], Cell[TextData[{ StyleBox[ "To decode the sequence of corrected code words, we want to apply the "], StyleBox["DecodeH74", FontSlant->"Italic"], StyleBox[ " linear transformation to each corrected code word in the list. "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "decodedinfobitwords =\n Map[DecodeH74,CorrectedCodeWords]"]], "Input",\ CellLabel->"In[146]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0, 0, 1, 1}, {1, 0, 0, 1}, { 1, 1, 1, 0}, {0, 0, 0, 1}, {1, 1, 0, 1}, {1, 0, 1, 1}, {1, 1, 0, 0}, { 0, 0, 1, 1}, {0, 1, 1, 0}, {0, 1, 1, 0}, {0, 1, 0, 1}, {0, 1, 0, 0}, { 0, 0, 1, 0}}\)], "Output", CellLabel->"Out[146]="] }, Open ]], Cell[TextData[StyleBox[ "Step 2. Convert the decoded information bit words to decoded bitstream."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "decodedbitstream = \n Flatten[decodedinfobitwords]"]], "Input", CellLabel->"In[147]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0}\)], "Output", CellLabel->"Out[147]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Convert from bitstream back to text"]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "We now convert the decoded bitstream back to text. If we've sucessfully \ corrected all errors, then the decoded text and the original text will be the \ same."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Step 1. Convert bitstream to a sequence of words "]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Convert the bitstream back to the sequence of binary vectors of the \ appropriate length."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "decodedbinary = \n BitstreamToBinary[decodedbitstream, 7]"]], "Input", CellLabel->"In[148]:=", AspectRatioFixed->True], Cell[BoxData[ \({{1, 0, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 1, 1}, { 1, 1, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0}, { 1, 1, 0, 1, 1, 0, 0}, {1, 1, 0, 0, 1, 0, 1}, {0, 1, 0, 0, 0, 0, 1}} \)], "Output", CellLabel->"Out[148]="] }, Open ]], Cell[TextData[StyleBox[ "Step 2. Convert the binary vectors back to text."]], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "decodedtext = \n BinaryToText[decodedbinary, 7]"]], "Input", CellLabel->"In[149]:=", AspectRatioFixed->True], Cell[BoxData[ \("A sample!"\)], "Output", CellLabel->"Out[149]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Try it yourself"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "To get a much better feel for what a Hamming(7,4) code can do, we want you \ to play around with the process of converting text to bitstream, coding the \ bitstream, adding error, and attempting to decode the received bitstream back \ to text. "]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "The commands that you need are organized into cells that correspond to the \ major steps in the coding/decoding process."]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Convert text to coded bitstream."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "You can change the text string from \"Fill in the blank?\" if you want to \ by highlighting the Fill in the blank? and typing the desired text over it. \ The text string "], StyleBox["must", FontSlant->"Italic"], StyleBox[" be inside the quote marks."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[textsring,binary, bitstream, \n infobitwords, codewords, \n \ codedbitstream];\n\ntextstring = \"Fill in the blank?\"\n\nbinary = \ TextToBinary[textstring, 7];\nbitstream = Flatten[binary];\ninfobitwords = \ InfoBits[bitstream, 4];\ncodewords = Map[Ham74, infobitwords];\n\ codedbitstream = Flatten[codewords]"]], "Input", CellLabel->"In[150]:=", AspectRatioFixed->True], Cell[BoxData[ \("Fill in the blank?"\)], "Output", CellLabel->"Out[151]="], Cell[BoxData[ \({1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0}\)], "Output", CellLabel->"Out[156]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Add error and see what is received ."]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Experiment with adding more than one error by changing the value of "], StyleBox["n", FontSlant->"Italic"], StyleBox[" in the next cell. "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Clear[n,receivedbitstream];\nn = 1;\nreceivedbitstream = \ AddErrors[codedbitstream, n]"]], "Input", CellLabel->"In[157]:=", AspectRatioFixed->True], Cell[BoxData[ \({1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0}\)], "Output", CellLabel->"Out[159]="] }, Open ]], Cell[TextData[StyleBox[ "To see the pattern of where the errors occured, look at the difference of \ the coded bitstream and the received bitstream. 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Places \ where 2 or more errors occur very closeto each other\[LongDash]say within 3 \ places of each other\[LongDash]are called \"error bursts.\" Use \ expermentation to determine what happens with error bursts."]], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Detect and correct the errors. 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