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Part 2 Linear block codes

1.14 Definition and Generator Matrix

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1.15 Coding Scheme

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1.16 RRE Generator Matrix

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1.17 Systematic Code

First definition:

Second definition:

The k information symbols appear in the first k positions of any

codeword.

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1.18 Dual Code and Parity Check Matrix

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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 def in i t ion of par i ty check matr ix: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 andparity check matrices with given parameters ? See additional

materials.

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The syndrome decoding is a kind of minimum

distance decoding.

Using syndrome decoding, for given y and then itssyndrome 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, thecomplexity for searching the closest codeword x, is

also qk

Then, use or do not use syndrome decoding ?

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1.20 Hamming code

Binary Hamming Code

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Nonbinary Hamming Code

Counting the number of binary Hamming codes and q-ary Hamming

codes with given parameters. See additional materials.