Abstract
We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call visible rank. The locality constraints of a linear code are stipulated by a matrix H of ⋆’s and 0’s (which we call a "stencil"), whose rows correspond to the local parity checks (with the ⋆’s indicating the support of the check). The visible rank of H is the largest r for which there is a r × r submatrix in H with a unique generalized diagonal of ⋆’s. The visible rank yields a fieldindependent combinatorial lower bound on the rank of H and thus the codimension of the code.
We point out connections of the visible rank to other notions in the literature such as unique restricted graph matchings, matroids, spanoids, and minrank. In particular, we prove a ranknullity type theorem relating visible rank to the rank of an associated construct called symmetric spanoid, which was introduced by Dvir, Gopi, Gu, and Wigderson [Zeev Dvir et al., 2020]. Using this connection and a construction of appropriate stencils, we answer a question posed in [Zeev Dvir et al., 2020] and demonstrate that symmetric spanoid rank cannot improve the currently best known Õ(n^{(q2)/(q1)}) upper bound on the dimension of qquery locally correctable codes (LCCs) of length n. This also pins down the efficacy of visible rank as a proxy for the dimension of LCCs.
We also study the tDisjoint Repair Group Property (tDRGP) of codes where each codeword symbol must belong to t disjoint check equations. It is known that linear codes with 2DRGP must have codimension Ω(√n) (which is matched by a simple product code construction). We show that there are stencils corresponding to 2DRGP with visible rank as small as O(log n). However, we show the second tensor of any 2DRGP stencil has visible rank Ω(n), thus recovering the Ω(√n) lower bound for 2DRGP. For qLCC, however, the k'th tensor power for k ⩽ n^{o(1)} is unable to improve the Õ(n^{(q2)/(q1)}) upper bound on the dimension of qLCCs by a polynomial factor.Inspired by this and as a notion of intrinsic interest, we define the notion of visible capacity of a stencil as the limiting visible rank of high tensor powers, analogous to Shannon capacity, and pose the question whether there can be large gaps between visible capacity and algebraic rank.
BibTeX  Entry
@InProceedings{alrabiah_et_al:LIPIcs.APPROX/RANDOM.2021.57,
author = {Alrabiah, Omar and Guruswami, Venkatesan},
title = {{Visible Rank and Codes with Locality}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
pages = {57:157:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772075},
ISSN = {18688969},
year = {2021},
volume = {207},
editor = {Wootters, Mary and Sanit\`{a}, Laura},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14750},
URN = {urn:nbn:de:0030drops147502},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.57},
annote = {Keywords: Visible Rank, Stencils, Locality, DRGP Codes, Locally Correctable Codes}
}
Keywords: 

Visible Rank, Stencils, Locality, DRGP Codes, Locally Correctable Codes 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021) 
Issue Date: 

2021 
Date of publication: 

15.09.2021 