Z_2Z_4-Additive Cyclic Codes Are Asymptotically Good

11/21/2019

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by   Yun Fan, et al.

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We construct a class of Z_2Z_4-additive cyclic codes generated by pairs of polynomials, study their algebraic structures, and obtain the generator matrix of any code in the class. Using a probabilistic method, we prove that, for any positive real number δ<1/3 such that the entropy at 3δ/2 is less than 1/2, the probability that the relative minimal distance of a random code in the class is greater than δ is almost 1; and the probability that the rate of the random code equals to 1/3 is also almost 1. As an obvious consequence, the Z_2Z_4-additive cyclic codes are asymptotically good.