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Hamming metric decoding of alternant codes over Galois rings

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2 Author(s)
Byrne, E. ; Dept. of Math., Univ. Coll. Cork, Ireland ; Fitzpatrick, P.

The standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence. We extend this technique to decode alternant codes over Galois rings. We consider the module M={(a, b): as≡b mod xr} of all solutions to the key equation where s is the syndrome polynomial and r, is the number of rows in a parity-check matrix for the code. In decoding we seek a particular solution (Σ, Ω)∈M which we prove can be found in a Grobner basis for M. We present an iterative algorithm which generates a Grobner basis modulo xk+1 from a given basis modulo xk. At the rth step, a Grobner basis for M is found, and the required solution recovered

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Information Theory, IEEE Transactions on  (Volume:48 ,  Issue: 3 )