ℤ4-Double Cyclic Codes Are Asymptotically Good

摘要

We construct a class of Z(4)-double cyclic codes generated by pairs of polynomials. Based on the probabilistic method, we prove the asymptotic properties of this class of codes: for any positive real number 0 < delta < 1 such that the 4-ary entropy at k+l/2 delta is less than 1/4, the rate of the random code is convergent to 1/k+l and the relative distance of the code is convergent to delta, where k and l are pairwise coprime positive odd integers. As a result, the Z(4)-double cyclic codes are asymptotically good.