Abstract
The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp‐Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23.
Highlights
Quadratic residue (QR) codes are cyclic, nominally half-rate codes, that are powerful with respect to their error-correction capabilities
This paper shows that one technique used to decode binary QR codes can be applied successfully to decode nonbinary QR codes
The method of determining unknown syndromes was first presented by He et al in [2] to decode the binary QR code of length 47 and subsequently to decode several other binary QR codes; see [6] and references therein
Summary
Quadratic residue (QR) codes are cyclic, nominally half-rate codes, that are powerful with respect to their error-correction capabilities. Decoding algorithms for certain nonbinary QR codes were proposed by Higgs and Humphreys in [3] and [4]. The main idea is to determine certain unknown syndromes in order to restore linearity to Newton’s identities. Once this is done, either the Peterson or the Berlekamp-Massey algorithm can be used to solve the identities. The method of determining unknown syndromes was first presented by He et al in [2] to decode the binary QR code of length 47 and subsequently to decode several other binary QR codes; see [6] and references therein. Error values can be found from the evaluator polynomial [7, p. 246] once the error locations are determined
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