On the number of error correcting codes

05/24/2022

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by   Dingding Dong, et al.

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We show that for a fixed q, the number of q-ary t-error correcting codes of length n is at most 2^(1 + o(1)) H_q(n,t) for all t ≤ (1 - q^-1)n - C_q√(n log n) (for sufficiently large constant C_q), where H_q(n, t) = q^n / V_q(n,t) is the Hamming bound and V_q(n,t) is the cardinality of the radius t Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for t = o(n^1/3 (log n)^-2/3).