On Permutation Groups of Cyclic Codes over Finite Fields
This work advances the theoretical understanding of permutation groups for cyclic codes, which is relevant for coding theory researchers studying weight distribution and decoding.
The paper determines the permutation groups of certain cyclic codes over finite fields with lengths hp, r^m p^n, and pq, where p, q, r are distinct primes, using matrix representations. For length pq, it provides the first results for codes with generator polynomials that are factors of x^{pq}-1 but not of x^p-1 or x^q-1.
The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic codes with very long lengths and special generator polynomials to those with prime lengths. Consequently, we mainly determine the permutation groups of certain cyclic codes over $\mathbb{F}_{r^α}$ with lengths $hp$, $r^mp^n$ and $pq$ and special generator polynomials where $h$ is a positive integer and $p$, $q$ and $r$ are distinct prime numbers. For length $pq$, we manage to provide the permutation groups of cyclic codes with generator polynomials $Q_{pq}(x)$(the $pq$-th cyclotomic polynomial) or others, which seems to be the first work about permutation groups of cyclic codes with generator polynomials that are factors of $x^{pq}-1$ but not factors of $x^p-1(\text{or }x^q-1)$.