Abstract-We introduce irregular product codes, a class of codes where each codeword is represented by a matrix and the entries in each row (column) of the matrix come from a component row (column) code. As opposed to standard product codes, we do not require that all component row codes nor all component column codes be the same. Relaxing this requirement can provide some additional attractive features such as allowing some regions of the codeword to be more error-resilient, providing a more refined spectrum of rates for finite lengths, and improved performance for some of these rates. We study these codes over erasure channels and prove that for any 0 < < 1, for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has asymptotic rate 1 − and can decode on erasure channels having erasure probability < (and having alphabet size equal to the alphabet size of the component MDS codes).
Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a "low-density parity check" (LDPC) code for it to be locally testable, but few LDPC codes are known to the locally testable, and even fewer classes of LDPC codes are known not to be locally testable. Indeed, most previous examples of codes that are not locally testable were also not LDPC. The only exception was in the work of Ben-Sasson et al. [2005] who showed that random LDPC codes are not locally testable. Random codes lack "structure" and in particular "symmetries" motivating the possibility that "symmetric LDPC" codes are locally testable, a question raised in the work of Alon et al. [2005]. If true such a result would capture many of the basic ingredients of known locally testable codes.In this work we rule out such a possibility by giving a highly symmetric ("2-transitive") family of LDPC codes that are not testable with constant number of queries. We do so by continuing the exploration of "affine-invariant codes" -codes where the coordinates of the words are associated with a finite field, and the code is invariant under affine transformations of the field. New to our study is the use of fields that have many subfields, and showing that such a setting allows sufficient richness to provide new obstacles to local testability, even in the presence of structure and symmetry.
Abstract-We analyze the second moment of the ripple size during the LT decoding process and prove that the standard deviation of the ripple size for an LT-code with length k is of the order of √ k. Together with a result by Karp et. al stating that the expectation of the ripple size is of the order of k [3], this gives bounds on the error probability of the LT decoder. We also give an analytic expression for the variance of the ripple size up to terms of constant order, and refine the expression in [3] for the expectation of the ripple size up to terms of the order of 1/k, thus providing a first step towards an analytic finite-length analysis of LT decoding.
Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field F q n to the subfield F q and include all properties that form an F q -vector space and are invariant under affine transformations of the domain. Almost all the known locally testable affine-invariant properties have so-called "single-orbit characterizations" -namely they are specified by a single local constraint on the property, and the "orbit" of this constraint, i.e., translations of this constraint induced by affine-invariance. Single-orbit characterizations by a local constraint are also known to imply local testability. Despite this prominent role in local testing for affine-invariant properties, single-orbit characterizations are not well-understood.In this work we show that properties with single-orbit characterizations are closed under "summation". Such a closure does not follow easily from definitions, and our proof uses some of the rich developing theory of affine-invariant properties. To complement this result, we also show that the property of being an n-variate low-degree polynomial over F q has a singleorbit characterization (even when the domain is viewed as F q n and so has very few affine transformations). This allows us to exploit known results on the single-orbit characterizability of "sparse" affine-invariant properties to show the following: The sum of any sparse affineinvariant property (properties satisfied by q O(n) -functions) with the set of degree d multivariate polynomials over F q has a single-orbit characterization (and is hence locally testable) when q is prime. Our result leads to the broadest known family of locally testable affine-invariant properties and gives rise to some intriguing questions/conjectures attempting to classify all locally testable affine-invariant properties.
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