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This article is not cryptoraphy related, it is about coding schemes for robust telecommunications.

I have a degree in Management Information Systems. I am trying to learn what a cyclic code is. This article has left an emotional impression, but I have not learned what I came here seeking. I am leaving to go elsewhere on the Internet to learn what a cyclic code is.

[email protected]

In the section "Algebraic structure", Wikipeida said:

An irreducible code is a cyclic code in which the code, as an ideal, is minimal in R, so that its generator is an irreducible polynomial.

But irreducible polynomial generate maximal ideal.

From J.H.van Lint's book "Introduction to Coding Theory" page 83, (6.1.6):

The cyclic code generated by ${\displaystyle f_{i}(x)}$ is called a maximal cyclic code(since it is a maximal ideal) and denote by ${\displaystyle M_{i}^{+}}$. The code generated by ${\displaystyle {\frac {x^{n}-1}{f_{i}(x)}}}$ is called a minimal cyclic code and denoted by ${\displaystyle M_{i}^{-}}$. Minimal cyclic codes are also called irreducible cyclic codes.

--Zxl.gzhu (talk) 09:33, 10 April 2010 (UTC)Reply

The section Cyclic code#For correcting errors makes the following claim:

The received word is a polynomial of degree ${\displaystyle n-1}$ given as ${\displaystyle v(x)=a(x)g(x)+e(x)}$

where ${\displaystyle e(x)}$ can have at most two nonzero coefficients corresponding to 2 errors.

I'm assuming this form of the received word is found by applying Euclidean division in ${\displaystyle GF(q)[x]}$. However, this does not guarantee that the remainder ${\displaystyle e(x)}$ has at most two nonzero coefficients (i.e. minimal Hamming weight) as the article claims. It instead guarantees that ${\displaystyle e(x)=0}$ or ${\displaystyle e(x)}$ has degree less than ${\displaystyle g(x)}$. One example where this does not give a remainder with at most two coefficients (for the Hamming code described in the article) is the received word ${\displaystyle v(x)=1+x+x^{2}}$.

In fact, the whole section feels a little bit off to me in terms of both confusing language and dubious math, but I'm not entirely sure how to fix it. Does anyone know of good references for this type of error correction? I think that might be a good place to start for cleaning up the article. Medissorize (talk) 15:17, 8 May 2025 (UTC)Reply