List Decoding Tensor Products and Interleaved Codes

Abstract

Abstract. We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes.• We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded.• We show that for every code, its list decoding radius remains unchanged under m-wise interleaving for an integer m. This generalizes a recent result of Dinur et al. [6], who proved such a result for interleaved Hadamard codes (equivalently, linear transformations).• Using the notion of generalized Hamming weights, we give better list size bounds for both tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of bounding the list size to bounding the number of close-by low-rank codewords. For decoding linear transformations, using rank-reduction together with other ideas, we obtain list size bounds that are tight over small fields.Our results give better bounds on the list decoding radius than what is obtained from the Johnson bound, and yield rather general families of codes decodable beyond the Johnson bound.

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“…Another approach comes from the list-decoding of tensor codes [4]. While the multivariate polynomial codes we are interested in are not tensor codes, they are subcodes of the code of polynomials with individual degree at most d. Using the algorithm of [4] for decoding tensor codes, we get an algorithm that can decode from a (1 − o(1))N errors when d = o(|S|), but fails to approach a constant fraction of the minimum distance when d approaches |S|.…”

Improved burst error correction via list decoding quasi-cyclic codes

“…In this section, we provide, for = 2, 3, classes of binary -quasi-cyclic codes for correcting -phased burst errors. We compare the performance of Algorithm 1 applied to these codes with list decoding of a BCH code over F 2 and with the (currently best) collaborative list decoding approach of Gopalan et al [8] when using -interleaved BCH codes over F 2 . We recall therefore the main theorem of Gopalan et al [8].…”

“…6). Furthermore, for = 2, 3, we identify classes of binary -quasi-cyclic codes whose dimensions are larger than those of binary -interleaved A alternant codes when decoded (up to the same radius as the quasi-cyclic codes) by the (currently best) collaborative list decoding approach of Gopalan et al [8].…”

List decoding of crisscross error patterns

“…Theorem 2 in Section II-C clearly follows from Theorem 3. [6] how to list decode each interleaved code (i.e, each row is a codeword of a constituent code and errors occur only in columns) up to the decoding radius of the constituent code with a slightly larger list size than the one of the constituent code.…”

“…It is not clear how to translate the approach from [6] to the cover metric. This is due to the fact that the cover of rows and columns of a matrix can change when adding further diagonals whereas the cover of only columns can only be extended, but no elements can be removed from the set of lines when adding further rows.…”