Representing Code over integer rings

asked 2023-07-31 18:09:21 +0200

vidyarthi gravatar image

updated 2023-08-16 00:40:45 +0200

Is there a method to deal with all codewords over the integer rings Z_n where n=p^k for a prime p? I know that the linear codes over general rings has yet to be implemented. But at least over prime power integer rings, is it somehow possible to get the codewords? By codeword, I mean a submodule of a free-module over the ring, in this case Z_n.

Specifically, my aim is come with a framework for computing the various parameters of codes over non-field integer rings. A more typical example would be this CCO paper on simplex code. In the paper, the preliminaries (section 2) and the first three theorems describe the simplex codes for the submodule of the free module Z_q^2. Similarly, the sections 3 and 4 give the definitions and generator matrices for unit macdonald and simplex unit macdonald codes over the free module Z_q^k where q and k are any integers. Any hints?Thanks beforehand.

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What is a codeword in or over $R=\Bbb Z/n$ (in the special case $n=p^k$)? What does it mean to "get" them? As a list? As an iterator? For which values of $n$, moderately big...?

dan_fulea gravatar imagedan_fulea ( 2023-08-01 01:33:07 +0200 )edit

@dan_fulea By codewords I mean a submodule over the ring [math]\mathbb{Z}_n[/math]. And yes, it is best to get them as a list. The main object is to study usual algebraic coding theory over these specific rings instead of the codes normally defined as vector spaces over fields.

vidyarthi gravatar imagevidyarthi ( 2023-08-01 11:28:42 +0200 )edit

Please use latex / mathjax to format mathematical expressions. So instead of [math]\mathbb{Z}_n[/math] simply $\mathbb{Z}_n$.

dan_fulea gravatar imagedan_fulea ( 2023-08-06 20:37:25 +0200 )edit

We still need a definition or a reference of a "code", what are "codewords" (maybe of some code) and what kind of operations should be performed with it? Modules, which are submodules of a given (finite) ring $R$ are called ideals of the ring. All what we want from such an ideal $J$ of $R$ is to be a subset of $R$, closed/stable w.r.t. the ring operation. Since our $R$ of interest is cyclic, generated by one element, each ideal is also generated by one element. There is not too much to search for inside such a structure, at least from a/my theoretical point of view. Do we have any explicit example, is there any article doing so.

dan_fulea gravatar imagedan_fulea ( 2023-08-06 20:45:26 +0200 )edit

@dan_fulea Modified the question. Added a reference. At least, if there was a workaround when q or n in this question is a prime power would be quite illuminating.

vidyarthi gravatar imagevidyarthi ( 2023-08-16 00:41:58 +0200 )edit