Abstract
This article explores two families of cyclic codes over Fq of length n denoted by Cn and Cn,1, which are generated by the n-th cyclotomic polynomial Qn(x) and the polynomial Qn(x)Q1(x), respectively. We find formulae for the distance of Cn and Cn,1 for each n>1 and conjecture formulae for the distance of their (Euclidean) duals. We prove the conjecture when n is a product of at most two distinct prime powers. Moreover, we show that all these codes are LCD codes, and several subfamilies are both r-optimal and d-optimal locally recoverable codes.
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