Optimal covering polynomial sets correcting three errors for binary cyclic codes

초록

The covering polynomial method is a generalization of error-trapping decoding and is a simple and effective way to decode cyclic codes. For cyclic codes of rate R < 2/tau, covering polynomials of a single term suffice to correct up to tau errors, and minimal sets of covering polynomials are known for various such codes. In this correspondence, the case of tau = 3 and of binary cyclic codes of rate R greater than or equal to 2/3 is investigated. Specifically, a closed-form specification is given for minimal covering polynomial sets for codes of rate 2/3 less than or equal to R < 11/15 for all sufficiently large code length n; the resulting number of covering polynomials is, if R = 2/3 + rho with rho > 0, equal to nrho + 2rootnrho + (1/2) logphi(n/rho) + O(l), where phi = (1 + root5)/2. For all codes correcting up to three errors, the number of covering polynomials is at least nrho + 2rootnrho + O(log n); covering polynomial sets achieving this bound (and thus within O(log n) of the minimum) are presented in closed-form specifications for rates in the range 11/15 less than or equal to R < 3/4.

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