A Combination of Testability and Decodability by Tensor Products

TL;DR

This paper proves that tensor products of any linear codes are locally testable, enabling the construction of efficient, highly reliable codes over any field, including binary, with optimal parameters.

Contribution

It improves previous results by showing tensor products are locally testable for all linear codes, not just those with large distance, enabling broader code constructions.

Findings

Tensor products of any linear codes are locally testable.

Constructs codes with constant rate, distance, and sublinear query complexity.

Enables linear-time encoding and decoding over any field.

Abstract

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as a black box) to present a combinatorial construction of locally testable codes that match best known parameters. As a consequence, this construction was obtained over sufficiently large fields. In this paper we improve the result of Ben-Sasson and Sudan and show that for \emph{any} linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field. Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties: have constant rate and constant relative distance; have blocklength $n$ and testable with $n^{\epsilon}$ queries, for any constant $\epsilon > 0$; linear time encodable and linear-time decodable from a constant fraction of errors. Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary regarding the list-decodable codes.