Part 2 Linear block codes 1.14 Definition and Generator Matrix

1.15 Coding Scheme

1.16 RRE Generator Matrix

1.17 Systematic Code First definition: Second definition: The k information symbols appear in the first k positions of any codeword.

1.18 Dual Code and Parity Check Matrix

The dual code of an [n, k] linear with generator matrix G=[Ik, A], is an [n, n-k] linear code with parity check matrix H=[-AT, In-k]. Another definition of parity check matrix: Let G and H be two matrices with full row rank. G is a generator of a linear block code. Then, HGT=O if and only if H is a parity check matrix. How to count the number of generator matrices and parity check matrices with given parameters ? See additional materials.

From G to H: RRE G’ Use column permutation to ensure G’’= [Ik, A] H’’=[-AT, In-k] (corresponding to G’’) Use inverse column permutation on H’’, we get H’ (corresponding to G’) H=H’ (corresponding to G)

1.19 Syndrome decoding

The syndrome decoding is a kind of minimum distance decoding. Using syndrome decoding, for given y and then its syndrome s, the complexity for searching z with the minimal weight in the solutions of s=HZT, is qk Without using syndrome decoding, for given y, the complexity for searching the closest codeword x, is also qk Then, use or do not use syndrome decoding ?

1.20 Hamming code Binary Hamming Code

Nonbinary Hamming Code Counting the number of binary Hamming codes and q-ary Hamming codes with given parameters. See additional materials.