On a Group Under Which Symmetric Reed-Muller Codes are Invariant
Sibel Kurt Toplu, Talha Arikan, Pinar AydoğDu, OğUz Yayla
TL;DR
This work investigates which linear automorphisms leave symmetric Reed-Muller codes invariant, focusing on SRM codes defined via $E_q(n,r)$ and the evaluation set $\Omega_q(n)$. The authors derive explicit invariant subgroups for $n=2$ and $n=3$, identifying a GL(2,q) subgroup $M$ and a GL(3,q) subgroup $K$ that preserve SRM codes for appropriate parameter ranges, and they provide concrete examples to illustrate the results. They further show these invariant groups sit inside the affine group, suggesting an affine-algebraic structure governing SRM symmetries, and they propose a general conjectural subgroup $M = \{ P((b-a)I_n + aJ_n) \}$ for arbitrary $n$ with $q>r>\frac{n(n-1)}{2}$, while acknowledging that a complete automorphism characterization for all $n$ remains open. The findings advance our understanding of SRM symmetries and lay groundwork for exploiting these automorphisms in decoding and code-design contexts, while highlighting key open problems in full automorphism group determination. Overall, the paper narrows the gap between known RM/GRM invariances and the tailored symmetry properties of SRM codes, guiding future work on generalized invariance for higher dimensions.
Abstract
The Reed-Muller codes are a family of error-correcting codes that have been widely studied in coding theory. In 2020, Wei Yan and Sian-Jheng Lin introduced a variant of Reed-Muller codes so called symmetric Reed-Muller codes. We investigate linear maps of the automorphism group of symmetric Reed-Muller codes and show that the set of these linear maps forms a subgroup of the general linear group, which is the automorphism group of punctured Reed-Muller codes. We provide a method to determine all the automorphisms in this subgroup explicitly for some special cases.