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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.