Code Automorphisms and Permutation Decoding of Certain Reed–Solomon Binary Images

We consider primitive Reed-Solomon (RS) codes over the field F_2m of length n=2^m-1. Building on Lacan ´s results for the case of binary extension fields, we show that the binary images of certain two-parity symbol RS [n, n-2, 3] code, have a code automorphism subgroup related to the general linear group GL(m, 2). For these codes, we obtain a code automorphism subgroup of order m! GL(m,2). An explicit algorithm is given to compute a code automorphism (if it exists), that sends a particular choice of m binary positions, into binary positions that correspond to a single symbol of the RS code. If one such automorphism exists for a particular choice of m binary symbol positions, we show that there are at least m! of them. Computationally efficient permutation decoders are designed for the two-parity symbol RS [n, n-2, 3] codes. Simulation results are shown for the additive white Gaussian noise (AWGN) channel. For the finite fields F₂₃ and F₂₄, we go on to derive subgroups of code automorphisms, belonging to binary images of certain RS codes that have three-parity symbols. A table of code automorphism subgroup orders, computed using the Groups, Algorithms, and Programming (GAP) software, is tabulated for the fields F₂₃, F₂₄ , and F₂₅.