On the Bounds of Certain Maximal Linear Codes in a Projective Space

DOI10.1109/TIT.2015.2449308zbMATH Open1359.94731arXiv1410.2725MaRDI QIDQ2977289

Author name not available (Why is that?)

Publication date: 28 April 2017

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: The set of all subspaces of $mathbb{F}_q^n$ is denoted by $mathbb{P}_q(n)$. The subspace distance $d_S(X,Y)=dim(X)+dim(Y)-2dim(XcapY)$ defined on $mathbb{P}_q(n)$ turns it into a natural coding space for error correction in random network coding. A subset of $mathbb{P}_q(n)$ is called a code and the subspaces that belong to the code are called codewords. Motivated by classical coding theory, a linear coding structure can be imposed on a subset of $mathbb{P}_q(n)$. Braun, Etzion and Vardy conjectured that the largest cardinality of a linear code, that contains $mathbb{F}_q^n$, is $2^n$. In this paper, we prove this conjecture and characterize the maximal linear codes that contain $mathbb{F}_q^n$.

Full work available at URL: https://arxiv.org/abs/1410.2725

Mathematics Subject Classification ID

Linear codes (general theory) (94B05)

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