On a group under which symmetric Reed-Muller codes are invariant

The Reed-Muller codes are a family of error-correcting codes that have been widely studied in coding theory. In 2020, Yan and Lin introduced a variant of Reed-Muller codes 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.