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

• Published in: IEEE transactions on information theory Vol. 56; no. 10; pp. 5253 - 5273
• Main Authors: Lim, Fabian, Fossorier, Marc, Kavčić, Aleksandar
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Automorphism group; Automorphisms; AWGN; binary images; Block codes; Channels; Codes; Coding, codes; Decoding; Exact sciences and technology; Finite element analysis; finite fields; Indexes; Information theory; Information, signal and communications theory; Materials; Normal distribution; permutation decoding; Permutations; Polynomials; Reed-Solomon (RS) codes; Signal and communications theory; Simulation; Software; Subgroups; Symbols; Telecommunications and information theory; Reed Solomon code; Automorphism; Finite field; AWGN; Programming; Reed-Solomon (RS) codes; Decoding; Automorphism group; Algorithm; AWGN channels; binary images; finite fields; Simulation; Binary image; permutation decoding
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2010.2059633

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 F 23 and F 24 , 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 F 23 , F 24 , and F 25 .