ℤ₄-Double Cyclic Codes Are Asymptotically Good

Abstract

We construct a class of Z4-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 <; δ <; 1 such that the 4-ary entropy at k+t/2 δ is less than 14, the rate of the random code is convergent to 1/k+t and the relative distance of the code is convergent to δ, where k and l are pairwise coprime positive odd integers. As a result, the Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -double cyclic codes are asymptotically good.

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