Overview
This study focuses on determining the permutation groups of cyclic codes over finite fields. Specifically, the research addresses cyclic codes over $\mathbb{F}_{r^\alpha}$ with particular lengths and generator polynomials. The methodology involves employing two distinct matrix representations to establish a relationship between cyclic codes of long lengths and those of prime lengths.
Research Context
Permutation groups of cyclic codes are recognized as having wide applicability in several areas related to coding theory. These applications include their utility in determining the weight distribution of codes and in decoding theory, among other areas. The current work contributes to this field by providing specific determinations for a class of cyclic codes.
Approach
The research approach leverages two distinct matrix representations. These representations serve as a mechanism to connect cyclic codes characterized by very long lengths and special generator polynomials to cyclic codes possessing prime lengths. This approach facilitates the determination of permutation groups for codes that otherwise might be more challenging to analyze directly.
Findings
- The study concentrated on determining the permutation groups of certain cyclic codes over $\mathbb{F}_{r^\alpha}$.
- The codes examined had specific lengths, denoted as $hp$, $r^\mp^n$, and $pq$. In these lengths, $h$ is defined as a positive integer, while $p$, $q$, and $r$ are distinct prime numbers.
- The cyclic codes under investigation also featured special generator polynomials.
- For cyclic codes of length $pq$, the research successfully managed to provide the permutation groups.
- This determination for length $pq$ included codes with generator polynomials such as $Q_{pq}(x)$, which represents the $pq$-th cyclotomic polynomial.
- The study also addressed other generator polynomials for codes of length $pq$, specifically those that are factors of $x^{pq}-1$ but are not factors of $x^p-1$ or $x^q-1$. This particular aspect is noted as a novel contribution in the context of permutation groups of cyclic codes.
Why This Matters
Permutation groups have established utility in determining the weight distribution of codes and in decoding theory. The findings of this research, by providing methods and specific determinations for these groups for various cyclic codes, contribute to the foundational understanding and potential advancements in these applicable areas.