II. Linear Block Codes

Weight Distribution of a Block Code For an (n.k) block code: Let Ai be the number of codewords of weight i in C. The numbers A0, A1, …, An are called the Weight Distribution Example: For the (7,4) code shown, A0 = A7 = 1, A1 = A2 = A5 = A6= 0 A3 = A4 = 7 Note that: ∑Ai=16=24 (Number of valid code words)

Weight Distribution and Probability of Detecting an Error Pattern For an (n,k) linear code: Given that the bit error probability of the physical channel is p. The probability that an error pattern of weight j occurs is pj(1-p)n-j In Total, there are nCj error patterns that have j erroneous bits. ONLY Aj of those are NOT DETECTABLE because they represent valid codewords Probability of not detecting an error pattern (Pu(E)) is: Example: For the (7,4) code in the previous slide, Pu(E)=7p3(1-p)4 + 7p4(1-p)3 + p7

Weight Distribution of Coset Leaders Let αi denote the number of coset leaders of weight i. Then the numbers α0,α1,…, αn are the weight distribution of the coset leaders Probability of Correct Decoding of an error pattern Ps(E) PS(E)=∑ αi pi(1-p)(n-i) where is the bit error probability of the channel Probability of False Decoding of an error pattern P(E) P (E)=1-PS(E)

Hamming Linear Block Codes

Hamming Codes For any positive integer m≥3, there exists a Hamming code such that: Code Length: n = 2m-1 No. of information symbols: k = n-m =2m-m-1 No. of parity check symbols: n-k = m Error correcting capability: t = 1 (i.e., dmin=3)

Parity Check Matrix, Generator Matrix and dmin of Hamming Codes H contains all of nonzero m-tuples as its columns. In the systematic form: m columns of weight 1 2m-m-1 columns of weight >1 Parity Check Matrix Generator Matrix No two columns are identical dmin>2 The sum of any two columns must be a third one dmin=3 Hamming Codes can correct all single errors or detect all double errors

Example: (15,11) Hamming Codes Code Length: n = 24-1 = 15 No. of information symbols: k = 15-4 = 11 No. of parity check symbols: 15-11 = 4 Error correcting capability: dmin = 3 4 columns of weight 1 11 columns of weight >1

Hamming Codes are Perfect Codes Consider the standard array for a Hamming Code The code has 2m cosets The number of (2m-1)-tuples of weight 1 is 2m-1 The 0 vector and the (2m-1)-tuples of weight 1 form all the coset leaders of the standard array In such configuration for the standard array, the Hamming code corrects ONLY error patterns with single error. A Perfect Code is a t-error correcting code where its standard array has all error patterns of t or fewer errors and no others as coset leaders

The (15,11) Hamming Code is a Perfect Code Number of cosets = 2n-k= 215-11 = 16 Number of error patterns of weight 1 = 15 If the coset leaders are chosen to be the 0 vector and all the error patterns of weight 1. The (15,11) code corrects ONLY error patterns with single error

Shortened Hamming Codes It is possible to delete l columns from the parity check matrix of H of a Hamming code. This results is H’ of dimensions m×(2m-l-1) which is the parity check matrix of a shortened Hamming code Code Length: n = 2m-l-1 No. of information symbols: k = n-m = 2m-m-l-1 No. of parity check symbols: n-k = m Error correcting capability: dmin≥3

Single Error Correction and Double Error Detection For a code with minimum Hamming distance dmin: If the code could correct λ and detect l then dmin≥ λ+l+1 For a shortened Hamming code to achieve single error correction (λ=1) and double error detection (l=2), It is essential that dmin≥4 To achieve dmin=4 it is essential to delete columns properly from H of the original Hamming code Example Delete from Q all even weight columns H’ is of dimension m×2m-1 Why is it that dmin=4? No two columns are identical  dmin>2 Adding two m-tuples of odd weight results in an even weight m-tuple  dmin>3 Adding three m-tuples of odd weight results in an odd weight m-tuple  dmin=4 2m-1-m odd weight columns

Single Error Correction and Double Error Detection (Decoding Scheme) If the syndrome s is zero, assume no error has occurred. If s is nonzero and has odd weight, assume single error has occurred. Add the error pattern corresponding to the syndrome. If s is nonzero and has even weight, an uncorrectable pattern has been detected

Example: (8,4) Shortened Hamming Code Parity Check Matrix for (15,11) Hamming Code delete all even weight columns

Example: (8,4) Shortened Hamming Code syndrome s=e.H’T Notice that If an error pattern has a single error, the syndrome has odd weight If an error pattern has a double error, the syndrome has even weight