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Sequences of linear codes where the rate times distance grows rapidly

Alizadeh, Faezeh ; Glasby, S P ; Praeger, Cheryl E

arXiv.org, 2021-10

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Título:
Sequences of linear codes where the rate times distance grows rapidly
Autor: Alizadeh, Faezeh ; Glasby, S P ; Praeger, Cheryl E
Assuntos: Codes ; Computer Science - Information Theory ; Linear codes ; Mathematics - Combinatorics ; Mathematics - Information Theory
É parte de: arXiv.org, 2021-10
Descrição: For a linear code \(C\) of length \(n\) with dimension \(k\) and minimum distance \(d\), it is desirable that the quantity \(kd/n\) is large. Given an arbitrary field \(\mathbb{F}\), we introduce a novel, but elementary, construction that produces a recursively defined sequence of \(\mathbb{F}\)-linear codes \(C_1,C_2, C_3, \dots\) with parameters \([n_i, k_i, d_i]\) such that \(k_id_i/n_i\) grows quickly in the sense that \(k_id_i/n_i>\sqrt{k_i}-1>2i-1\). Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is \(\mathbb{F}=\mathbb{F}_2\) and \(k_i d_i/n_i\) is asymptotic to \(3n_i^{c}/\sqrt{\pi\log_2(n_i)}\) where \(c=\log_2(3/2)\approx 0.585\).
Editor: Ithaca: Cornell University Library, arXiv.org
Idioma: Inglês

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