Optimal covering polynomial sets correcting three errors for binary cyclic codes

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/τ, covering polynomials of a single term suffice to correct up to τ errors, and minimal sets of covering polynomials are known for various such codes. In this article, the case of τ=3 and of binary cyclic codes of rate R⩾2/3 is investigated. Specifically, a closed-form specification is given for minimal covering polynomial sets for codes of rate 2/3⩽R<11/15 for all sufficiently large code length n; the resulting number of covering polynomials is, if R=2/3+ρ with ρ>0, equal to nρ+2V√nρ+(1/2) log_φ(n/ρ)+O(1), where φ=(1+√5)/2. For all codes correcting up to three errors, the number of covering polynomials is at least nρ+2√nρ+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⩽R<3/4