Some Recent Results on Local Testing of Sparse Linear Codes

Abstract

We study the local testability of linear codes. Our approach is based on a reformulation of this question in the language of tolerant linearity testing under a non-uniform distribution. We then study the question of linearity testing under non-uniform distributions directly, and give a sufficient criterion for linearity to be tolerantly testable under a given distribution. We show that several natural classes of distributions satisfy this criterion (such as product distributions and low Fourier-bias distributions), thus showing that linearity is tolerantly testable under these distributions. This in turn implies that the corresponding codes are locally testable.

For the case of random sparse linear codes, we show the testability and decodability of such codes in the presence of very high noise rates. More precisely, we show that any linear code in $\mathbb{F}_2^n$ which is:

• sparse (i.e., has only poly(n) codewords)

• unbiased (i.e., each nonzero codeword has Hamming weight in (1/2 − n ^− γ, 1/2 + n ^− γ) for some constant γ> 0)

can be locally tested and locally list decoded from (1/2 − ε)-fraction errors using only $poly(\frac{1}{\epsilon})$ queries to the received word. This simultaneously simplifies and strengthens a result of Kaufman and Sudan, who gave a local tester and local (unique) decoder for such codes from some constant fraction of errors. For the case of Dual BCH codes, our algorithms can also be made to run in sublinear time.

Building on the methods used for the local algorithms, we also give sub-exponential time algorithms for list-decoding arbitrary unbiased (but not necessarily sparse) linear codes in the high-error regime.

The current extended abstract is a summary of some of the results that appeared in “Tolerant linearity testing and locally testable codes” that appeared in RANDOM 2009 and “Local list-decoding and testing of random linear codes from high-error” that will appear in STOC 2010.