Cyclic code

[_luca.barletta]

Prove that a binary cyclic code of length $N$, with minimum Hamming distance $d>=3$, cannot have words containing $0

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vl4dster

14 lug 2007, 20:34

$C$ is a cyclic code iff
it is a linear code such that if $c = (c_1, ..., c_N) \in C$ then $(c_N, c_1, ..., c_{N-1})$ is also in $C$.
Informally, it is required that the codewords are "closed" under cyclic permutations of $N$ elements without fixpoints.

Suppose for the sake of contradiction that there exists a codeword such as
$c=(c_0, ..., c_{N-1})$ with $c_i,\cdots ,c_{i+n-1 mod(N)) = 1$ for some $0 and $c_{j}=0$ in all the remaining positions.
Then, due to the fact that we are in $GF(2)$,
if we permute $c$ by the cyclic permutation $(0, 1, ..., N-1)$ or by a
transposition of the 2 coordinates $i$ and $i+n mod(N)$ we get the same codeword, $c_2$, which is also in $C$.
But then the hamming distance is 2, so this is a contradiction to the hypothesis $d_{"min"} \geq 3$.

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_luca.barletta

16 lug 2007, 14:16

Ok vl4d, good explanation.
Now consider a binary BCH code, under which condition/s the all 1s word is a codeword?

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[vl4dster]

$C$ is a cyclic code iff
it is a linear code such that if $c = (c_1, ..., c_N) \in C$ then $(c_N, c_1, ..., c_{N-1})$ is also in $C$.
Informally, it is required that the codewords are "closed" under cyclic permutations of $N$ elements without fixpoints.

Suppose for the sake of contradiction that there exists a codeword such as
$c=(c_0, ..., c_{N-1})$ with $c_i,\cdots ,c_{i+n-1 mod(N)) = 1$ for some $0 and $c_{j}=0$ in all the remaining positions.
Then, due to the fact that we are in $GF(2)$,
if we permute $c$ by the cyclic permutation $(0, 1, ..., N-1)$ or by a
transposition of the 2 coordinates $i$ and $i+n mod(N)$ we get the same codeword, $c_2$, which is also in $C$.
But then the hamming distance is 2, so this is a contradiction to the hypothesis $d_{"min"} \geq 3$.

[_luca.barletta]

Ok vl4d, good explanation.
Now consider a binary BCH code, under which condition/s the all 1s word is a codeword?