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- W2953156772 abstract "Let $mathcal{C} = (C_1, C_2, ldots)$ be a sequence of codes such that each $C_i$ is a linear $[n_i,k_i,d_i]$-code over some fixed finite field $mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the dimension, and $d_i$ is the minimum distance. We say that $mathcal{C}$ is asymptotically good if, for some $varepsilon > 0$ and for all $i$, $n_i geq i$, $k_i/n_i geq varepsilon$, and $d_i/n_i geq varepsilon$. Sequences of asymptotically good codes exist. We prove that if $mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n geq 1$), closed under puncturing and shortening, and if $mathcal{C}$ contains an asymptotically good sequence, then $mathcal{C}$ must contain all GF$(p)$-linear codes. Our proof relies on a powerful new result from matroid structure theory." @default.
- W2953156772 created "2019-06-27" @default.
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- W2953156772 date "2014-04-30" @default.
- W2953156772 modified "2023-09-27" @default.
- W2953156772 title "On the existence of asymptotically good linear codes in minor-closed classes" @default.
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- W2953156772 hasPublicationYear "2014" @default.
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