Properties of Linear Block CodesSaravanan Vijayakumaransarva@ee.iitb.ac.inDepartment of Electrical EngineeringIndian Institute of Technology BombayJuly 31, 20151 / 17Minimum Distance of a Linear Block CodeDenitionThe minimum distance of a block code C is dened asdmin= minx,yC,x=yd(x, y)TheoremThe minimum distance of a linear block code is equal to theminimum weight of its nonzero codewordsProof.dmin= min_wt(x + y)x, y C, x = y_= min_wt(v)v C, v = 0_2 / 17ExampleFind the minimum distance of a linear block with parity checkmatrixH =__1 0 0 1 0 1 10 1 0 1 1 1 00 0 1 0 1 1 1__TheoremLet C be a binary linear block code with parity check matrix H.There exists a codeword of weight w in Cthere exist wcolumns in H which sum to the zero vector.CorollaryIf no w 1 or fewer columns of H sum to 0, the code hasminimum distance at least w.CorollaryThe minimum distance of C is the equal to the smallest numberof columns of H which sum to 0.3 / 17Singleton BoundLet C be an (n, k) binary block code with minimum distancedmin.dmin n k+ 1Proof.Suppose C is a linear block code.What is the rank of H?Suppose C is not a linear block code.Puncture the rst dmin 1 locations in each codeword.Can two punctured codewords be the same?4 / 17Error Detection using Linear Block CodesSuppose an (n, k, dmin) linear block code C is used forerror detectionLet x be the transmitted codeword and y is the receivedvectory = x + eThe receiver declares an error if y is not a codewordAny error pattern of weight dmin 1 or less will be detectedOf the 2n1 nonzero error patterns 2k1 are the sameas nonzero codewords in C 2k1 error patterns areundetectable and 2n2kare detectableLet Ai be the number of codewords of weight i in CProbability of undetected error over a BSC is given byPue=n

i =1Aipi(1 p)ni5 / 17ExampleFind the weight distribution of a linear block with parity checkmatrixH =__1 0 0 1 0 1 10 1 0 1 1 1 00 0 1 0 1 1 1__A0= 1, A7= 1, A1= 0, A2= 0, A3= 7, A4= 7, A5= 0, A6= 0Pue= 7p3(1 p)4+ 7p4(1 p)3+ p76 / 17Probability of Undetected ErrorPue= 7p3(1 p)4+ 7p4(1 p)3+ p7101102103101102103104105106107108pPue7p37 / 17The Standard ArrayLet C be an (n, k) linear block codeLet v1, v2, . . . , v2k be the codewords in C with v1= 0The standard array for C is constructed as follows1. Put the codewords viin the rst row starting with 02. Find a smallest weight vector e Fn2 not already in the array3. Put the vectors e + viin the next row starting with e4. Repeat steps 2 and 3 until all vectors in Fn2 appear in thearrayExample: G =_1 1 0 00 0 1 1_0000 1100 0011 11111000 0100 1011 01110010 1110 0001 11010110 1010 0101 10018 / 17Properties of the Standard ArrayEach row has 2kdistinct vectorsThe rows are disjointThere are 2nkrowsThe rows are called cosets of the code CThe rst vector in each row is called a coset leaderDecoding using the standard array Let 0, e2, e3, . . . , e2nk be the coset leaders Let Djbe the j th column of the standard arrayDj= {vj, e2 + vj, e3 + vj, . . . , e2nk + vj} Decode a vector which belongs to Djto vj Any error pattern equal to a coset leader is correctableEvery (n, k) linear block code can correct 2nkerrorpatterns9 / 17ExampleG =__0 1 1 1 0 01 0 1 0 1 01 1 0 0 0 1__000000 011100 101010 110001 110110 101101 011011 000111100000 111100 001010 010001 010110 001101 111011 100111010000 001100 111010 100001 100110 111101 001011 010111001000 010100 100010 111001 111110 100101 010011 001111000100 011000 101110 110101 110010 101001 011111 000011000010 011110 101000 110011 110100 101111 011001 000101000001 011101 101011 110000 110111 101100 011010 000110100100 111000 001110 010101 010010 001001 111111 100011The code has minimum distance 3It corrects all single-bit errors and one double-bit error10 / 17Syndrome DecodingAll vectors in the same row of the standard array have thesame syndromeVectors in different rows have different syndromesSteps in syndrome decoding Compute the syndrome y HTof the received vector y Find the coset leader eiwhose syndrome equals y HT Decode y into the codeword v = y + eiLet i be the number of coset leaders of weight i for CProbability of decoding error over a BSC is given byPe= 1 n

i =0ipi(1 p)ni11 / 17Probability of Decoding ErrorG =__0 1 1 1 0 01 0 1 0 1 01 1 0 0 0 1__Pe= 1 (1 p)66p(1 p)5p2(1 p)4101102103101102103104105pPe12 / 17Hamming BoundLet C be an (n, k) binary linear block code with minimumdistance dmin 2t + 1.2nk 1 +_n1_+_n2_+ +_nt_Proof.Does dmin 2t + 1 imply that all vectors of weight t or less arecoset leaders?Suppose wt(x) t and wt(y) t . Can x and y be in the samecoset?13 / 17MacWilliams IdentityLet C be an (n, k) binary linear block codeLet A0, A1, . . . , An be the weight distribution of CLet B0, B1, . . . , Bn be the weight distribution of CThe corresponding weight enumerators are given byA(z) = A0 + A1z + AnznB(z) = B0 + B1z + BnznThe MacWilliams identity states thatA(z) = 2(nk)(1 + z)nB_1 z1 + z_14 / 17ExampleH =__1 0 0 1 0 1 10 1 0 1 1 1 00 0 1 0 1 1 1__A(z) = 1 + 7z3+ 7z4+ z7B(z) = 1 + 7z423(1 + z)7B_1 z1 + z_= 23(1 + z)7_1 + 7_1 z1 + z_4_15 / 17Pue and A(z)Probability of undetected error over a BSC is given byPue=n

i =1Aipi(1 p)ni= (1 p)nn

i =1Ai_p1 p_i= (1 p)n_1 +n

i =0Ai_p1 p_i_= (1 p)n_A_p1 p_1_16 / 17Questions?17 / 17