Code automorphisms and permutation decoding of certain Reed-Solomon binary images

  • Authors:

  • Fabian Lim;Marc Fossorier;Aleksandar Kavčić

  • Affiliations:

  • Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI;ETIS, ENSEA, UCP, CNRS, UMR, Cergy Pontoise, France;Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

We consider primitive Reed-Solomon (RS) codes over the field F2m of length n = 2m - 1. Building on Lacan et al.'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! ċ |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.