18,820 research outputs found

    Combining subspace codes

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    In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size Aq(n,d;k)A_q(n, d; k) of codes whose codewords are kk-subspaces of Fqn\mathbb{F}_q^n with minimum subspace distance dd. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant--dimension subspace codes for many parameters, including Aq(10,4;5)A_q(10, 4; 5), Aq(12,4;4)A_q(12, 4; 4), Aq(12,6,6)A_q(12, 6, 6) and Aq(16,4;4)A_q(16, 4; 4).Comment: 17 pages; construction for A_(10,4;5) was flawe

    A new upper bound for subspace codes

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    It is shown that the maximum size A2(8,6;4)A_2(8,6;4) of a binary subspace code of packet length v=8v=8, minimum subspace distance d=4d=4, and constant dimension k=4k=4 is at most 272272. In Finite Geometry terms, the maximum number of solids in PG(7,2)\operatorname{PG}(7,2), mutually intersecting in at most a point, is at most 272272. Previously, the best known upper bound A2(8,6;4)289A_2(8,6;4)\le 289 was implied by the Johnson bound and the maximum size A2(7,6;3)=17A_2(7,6;3)=17 of partial plane spreads in PG(6,2)\operatorname{PG}(6,2). The result was obtained by combining the classification of subspace codes with parameters (7,17,6;3)2(7,17,6;3)_2 and (7,34,5;{3,4})2(7,34,5;\{3,4\})_2 with integer linear programming techniques. The classification of (7,33,5;{3,4})2(7,33,5;\{3,4\})_2 subspace codes is obtained as a byproduct.Comment: 9 page

    Linear-algebraic list decoding of folded Reed-Solomon codes

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    Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)} time algorithm to list decode appropriate folded RS codes of rate RR from a fraction 1-R-\eps of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices if one settles for a smaller decoding radius (but still enough for a statement of the above form). Here we give a simple linear-algebra based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in {\em quadratic} time. The theoretical drawback of folded RS codes are that both the decoding complexity and proven worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list size bound of O(1/\eps^2) which is quite close to the existential O(1/\eps) bound (however, the decoding complexity remains n^{\Omega(1/\eps)}). Our work highlights that constructing an explicit {\em subspace-evasive} subset that has small intersection with low-dimensional subspaces could lead to explicit codes with better list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on Computational Complexity (CCC), 201

    Evading Subspaces Over Large Fields and Explicit List-decodable Rank-metric Codes

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    We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC\u2713). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS\u2713) with subspace-evasive varieties due to Dvir and Lovett (STOC\u2712). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes
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