Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over \F_q, for many classes of $q$'s. This is a partial solution to a fifty year old open problem

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Asymmetric Quantum Codes: New Codes from Old

In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.Comment: Accepted for publication Quantum Information Processin

Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear error-correcting codes that have the "single local orbit" property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every "sparse" binary code whose coordinates are indexed by elements of F_{2^n} for prime n, and whose symmetry group includes the group of non-singular affine transformations of F_{2^n} has the single local orbit property. (A code is said to be "sparse" if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short (O(n)-bit, as opposed to the natural exp(n)-bit) description of a low-weight basis for BCH codes. The interest in the "single local orbit" property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates F_{2^n} for prime n are locally testable. If, in addition to n being prime, if 2^n-1 is also prime (i.e., 2^n-1 is a Mersenne prime), then we get that every sparse cyclic code also has the single local orbit. In particular this implies that BCH codes of Mersenne prime length are generated by a single low-weight codeword and its cyclic shifts