Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation

This paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient numerical technique for its evaluation is presented, involving the solution of a 4 x 4 system of polynomial equations. The expression is consistent with previous results, including the case where the normalized weight or stopping set size tends to zero. The spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree, a case which includes Tanner codes; for this case it is also shown how certain symmetry properties of the local weight distribution at the CNs induce a symmetry in the overall weight spectral shape function. Finally, using these new results, weight and stopping set size spectral shapes are evaluated for some example generalized and doubly-generalized LDPC code ensembles.Comment: 17 pages, 6 figures. To appear in IEEE Transactions on Information Theor

Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on $n$-vertex hyper-graphs: * Coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, polylog n colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a 'query doubling' method. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form.Comment: 25 page

Estimates on the Size of Symbol Weight Codes

The study of codes for powerlines communication has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the years. In this work we study a type of code called the bounded symbol weight codes which was first introduced by Versfeld et al. in 2005, and a related family of codes that we term constant symbol weight codes. We provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes. We also give direct and recursive constructions of codes for certain parameters.Comment: 14 pages, 4 figure