Visible Rank and Codes with Locality
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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 field-independent combinatorial lower bound on the rank of H and thus the co-dimension of the code. We prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called symmetric spanoid, which was introduced by Dvir, Gopi, Gu, and Wigderson <cit.>. Using this connection and a construction of appropriate stencils, we answer a question posed in <cit.> and demonstrate that symmetric spanoid rank cannot improve the currently best known O(n^(q-2)/(q-1)) upper bound on the dimension of q-query locally correctable codes (LCCs) of length n. We also study the t-Disjoint Repair Group Property (t-DRGP) of codes where each codeword symbol must belong to t disjoint check equations. It is known that linear 2-DRGP codes must have co-dimension Ω(√(n)). We show that there are stencils corresponding to 2-DRGP with visible rank as small as O(log n). However, we show the second tensor of any 2-DRGP stencil has visible rank Ω(n), thus recovering the Ω(√(n)) lower bound for 2-DRGP. For q-LCC, however, the k'th tensor power for k≤ n^o(1) is unable to improve the O(n^(q-2)/(q-1)) upper bound on the dimension of q-LCCs by a polynomial factor.
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