On finding the largest minimum distance of locally recoverable codes: A graph theory approach

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-31 DOI:10.1016/j.disc.2024.114298
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Abstract

The [n,k,r]-Locally recoverable codes (LRC) studied in this work are a well-studied family of [n,k] linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of [n,k,r]-LRCs, denoted by D(n,k,r). LMD can be approximated within an additive term of one—it is known that D(n,k,r) is equal to either d or d1, where d=nkkr+2. Moreover, for a range of parameters, it is known precisely whether the distance D(n,k,r) is d or d1. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.
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关于寻找局部可恢复编码的最大最小距离:图论方法
本文所研究的 [n,k,r]-Locally recoverable 编码(LRC)是一个经过深入研究的 [n,k] 线性编码系列,其中每个符号的值最多可以通过 r 个其他符号的线性组合来恢复。本文研究的是 LMD 问题,即找出 [n,k,r]-LRC 的最大可能最小距离,用 D(n,k,r) 表示。众所周知,D(n,k,r) 等于 d⁎ 或 d⁎-1,其中 d⁎=n-k-⌈kr⌉+2。此外,对于一系列参数来说,距离 D(n,k,r) 是 d⁎ 还是 d⁎-1,都是已知的。然而,尽管做了大量工作,这个问题仍然没有解决。在这项工作中,我们将 LMD 转换为图论中的等价简单问题。利用这种转换,我们证明了 LMD 实例的难度至少与计算高周长最大图的大小相当,而计算高周长最大图的大小是极值图论中的一个难题。这证明了 LMD 虽然可以在 1 的加法项内近似,但一般来说很难求解。作为一个积极的结果,得益于极值图论中的转换和现有成果,我们解决了一系列代码参数下的 LMD 问题,而这在以前是没有解决过的。
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
期刊最新文献
Extremal bounds for pattern avoidance in multidimensional 0-1 matrices Completely regular codes with covering radius 1 and the second eigenvalue in 3-dimensional Hamming graphs Disconnected forbidden pairs force supereulerian graphs to be hamiltonian On finding the largest minimum distance of locally recoverable codes: A graph theory approach Rigid frameworks with dilation constraints
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