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Hull dimension of optimal binary LRCs with availability.

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Title: Hull dimension of optimal binary LRCs with availability.
Authors: Mukhopadhyay, Deep, Bagchi, Satya
Source: Cryptography & Communications; Mar2026, Vol. 18 Issue 2, p571-589, 19p
Subject Terms: PARITY-check matrix, BIPARTITE graphs, LINEAR codes, PARAMETERIZATION, ERROR-correcting codes
Abstract: Locally repairable codes (LRCs) have gained substantial interest among the family of linear codes due to their theoretical and practical relevance, with various classes of LRCs being developed and studied in the literature. A crucial aspect of these different classes of LRCs is that they introduce additional parameters beyond the standard length and dimension of a linear code by classifying its parity structure with specific properties, known as LRC topology. This work implements the additional locality and availability parameters through the topology of systematic optimal single parity LRCs in studying their Euclidean hull dimension over the binary field. We first analyze the intrinsic properties of several optimal parametric classes of the considered LRC topology through its standard parity check matrix. Furthermore, by utilizing these characteristics, we determine the hull dimension of such optimal parametric classes of binary LRCs and establish bounds on it. Moreover, various infinite parametric classes of optimal binary LRCs are obtained that have fixed hull dimension based on their parameters and independent of their explicit code structures, highlighting an essential facet of optimal LRC topology in classifying systematic binary linear codes with different hull dimension using their additional parameters. Based on these results, we construct optimal binary LRCs with small hull dimension (zero and one) using biadjacency matrix of biregular bipartite graph. [ABSTRACT FROM AUTHOR]
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Abstract:Locally repairable codes (LRCs) have gained substantial interest among the family of linear codes due to their theoretical and practical relevance, with various classes of LRCs being developed and studied in the literature. A crucial aspect of these different classes of LRCs is that they introduce additional parameters beyond the standard length and dimension of a linear code by classifying its parity structure with specific properties, known as LRC topology. This work implements the additional locality and availability parameters through the topology of systematic optimal single parity LRCs in studying their Euclidean hull dimension over the binary field. We first analyze the intrinsic properties of several optimal parametric classes of the considered LRC topology through its standard parity check matrix. Furthermore, by utilizing these characteristics, we determine the hull dimension of such optimal parametric classes of binary LRCs and establish bounds on it. Moreover, various infinite parametric classes of optimal binary LRCs are obtained that have fixed hull dimension based on their parameters and independent of their explicit code structures, highlighting an essential facet of optimal LRC topology in classifying systematic binary linear codes with different hull dimension using their additional parameters. Based on these results, we construct optimal binary LRCs with small hull dimension (zero and one) using biadjacency matrix of biregular bipartite graph. [ABSTRACT FROM AUTHOR]
ISSN:19362447
DOI:10.1007/s12095-025-00862-y