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Revision #2 to TR21-119 | 20th October 2023 01:17

Visible Rank and Codes with Locality

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Revision #2
Authors: Omar Alrabiah, Venkatesan Guruswami
Accepted on: 20th October 2023 01:17
Downloads: 24
Keywords: 


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 \emph{visible rank}. The locality constraints of a linear code are stipulated by a matrix $H$ of $0$'s and $\star$'s (which we call a "stencil"), whose rows correspond to the local parity checks (with the $\star$'s indicating the support of the check). The visible rank of $H$ is the largest $r$ for which there is a $r \times r$ submatrix in $H$ with a unique generalized diagonal of $\star$'s. The visible rank yields a field-independent combinatorial lower bound on the rank of any matrix obtained by replacing the stars in $H$ by nonzero field elements and thus the co-dimension 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 min-rank. In particular, we prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called \emph{symmetric spanoid}, which was introduced by Dvir, Gopi, Gu, and Wigderson (2020). Using this connection and a construction of appropriate stencils, we answer a question posed by Dvir, Gopi, Gu, and Wigderson (2020) and demonstrate that the symmetric spanoid rank cannot improve the currently best known $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-query 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 $t$-Disjoint Repair Group Property ($t$-DRGP) of codes where each codeword symbol must belong to $t$ disjoint check equations. It is known that linear codes with $2$-DRGP must have co-dimension $\Omega(\sqrt{n})$ (which is matched by a simple product code construction). We show that there are stencils corresponding to $2$-DRGP codes with visible rank as small as $O(\log{n})$. However, we show that the second tensor power of any $2$-DRGP stencil has visible rank $\Omega(n)$, thus recovering the $\Omega(\sqrt{n})$ lower bound on the co-dimension of $2$-DRGP codes. More broadly, we show that visible ranks of tensor powers yield sharper lower bounds on the co-dimension of the code. For $q$-LCCs, however, the $k$'th tensor power for $k \le n^{o(1)}$ is unable to improve the $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-LCCs by a polynomial factor. Inspired by this and as a notion of intrinsic interest, we define the notion of \emph{visible capacity} of a stencil as the limiting visible rank of high tensor powers, analogous to Shannon capacity, and pose the question of whether there can be large gaps between visible capacity and algebraic rank.



Changes to previous version:

Modified the proofs of Lemma 7, Theorem 12, and Theorem 23; incorporated clarifications and minor corrections to several parts of the paper.


Revision #1 to TR21-119 | 29th August 2021 02:45

Visible Rank and Codes with Locality





Revision #1
Authors: Omar Alrabiah, Venkatesan Guruswami
Accepted on: 29th August 2021 02:45
Downloads: 234
Keywords: 


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 $\star$'s and $0$'s (which we call a "stencil"), whose rows correspond to the local parity checks (with the $\star$'s indicating the support of the check). The visible rank of $H$ is the largest $r$ for which there is a $r \times r$ submatrix in $H$ with a unique generalized diagonal of $\star$'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 point out connections of the visible rank to other notions in the literature such as unique restricted graph matchings, matroids, spanoids, and min-rank. In particular, 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 [DGGW20]. Using this connection and a construction of appropriate stencils, we answer a question posed in [DGGW20] and demonstrate that symmetric spanoid rank cannot improve the currently best known $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-query 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 $t$-Disjoint Repair Group Property ($t$-DRGP) of codes where each codeword symbol must belong to $t$ disjoint check equations. It is known that linear codes with $2$-DRGP must have co-dimension $\Omega(\sqrt{n})$ (which is matched by a simple product code construction). 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 $\Omega(n)$, thus recovering the $\Omega(\sqrt{n})$ lower bound for $2$-DRGP. For $q$-LCC, however, the $k$'th tensor power for $k\le n^{o(1)}$ is unable to improve the $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-LCCs 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.



Changes to previous version:

Fixed several typos and reworded the proof of Theorem 10.


Paper:

TR21-119 | 13th August 2021 17:03

Visible Rank and Codes with Locality





TR21-119
Authors: Omar Alrabiah, Venkatesan Guruswami
Publication: 13th August 2021 17:04
Downloads: 335
Keywords: 


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 $\star$'s and $0$'s (which we call a "stencil"), whose rows correspond to the local parity checks (with the $\star$'s indicating the support of the check). The visible rank of $H$ is the largest $r$ for which there is a $r \times r$ submatrix in $H$ with a unique generalized diagonal of $\star$'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 point out connections of the visible rank to other notions in the literature such as unique restricted graph matchings, matroids, spanoids, and min-rank. In particular, 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 [DGWW]. Using this connection and a construction of appropriate stencils, we answer a question posed in [DGGW] and demonstrate that symmetric spanoid rank cannot improve the currently best known $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-query 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 $t$-Disjoint Repair Group Property ($t$-DRGP) of codes where each codeword symbol must belong to $t$ disjoint check equations. It is known that linear codes with $2$-DRGP must have co-dimension $\Omega(\sqrt{n})$ (which is matched by a simple product code construction). 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 $\Omega(n)$, thus recovering the $\Omega(\sqrt{n})$ lower bound for $2$-DRGP. For $q$-LCC, however, the $k$'th tensor power for $k\le n^{o(1)}$ is unable to improve the $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-LCCs 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.



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