Optimal Encoding of Binary Cyclic Codes

Houshou CHEN

Publication
IEICE TRANSACTIONS on Communications   Vol.E89-B    No.12    pp.3280-3287
Publication Date: 2006/12/01
Online ISSN: 1745-1345
DOI: 10.1093/ietcom/e89-b.12.3280
Print ISSN: 0916-8516
Type of Manuscript: PAPER
Category: Fundamental Theories for Communications
Keyword: 
linear block codes,  binary cyclic codes,  bit error probability,  optimal encoding,  systematic generator matrix,  

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Summary: 
This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p→0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.