Symmetries of weight enumerators and applications to Reed-Muller codes

Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To appear in Advances in Mathematics of Communication

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

Switching Quantum Dynamics for Fast Stabilization

Control strategies for dissipative preparation of target quantum states, both pure and mixed, and subspaces are obtained by switching between a set of available semigroup generators. We show that the class of problems of interest can be recast, from a control--theoretic perspective, into a switched-stabilization problem for linear dynamics. This is attained by a suitable affine transformation of the coherence-vector representation. In particular, we propose and compare stabilizing time-based and state-based switching rules for entangled state preparation, showing that the latter not only ensure faster convergence with respect to non-switching methods, but can designed so that they retain robustness with respect to initialization, as long as the target is a pure state or a subspace.Comment: 15 pages, 4 figure

Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

We present $O(m^3)$ algorithms for specifying the support of minimum-weight words of extended binary BCH codes of length $n=2^m$ and designed distance $d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}$ for some values of $m,i,s$, where $m$ may grow to infinity. The support is specified as the sum of two sets: a set of $2^{2i-1}-2^{i-1}$ elements, and a subspace of dimension $m-2i-s$, specified by a basis. In some detail, for designed distance $6\cdot 2^j$, we have a deterministic algorithm for even $m\geq 4$, and a probabilistic algorithm with success probability $1-O(2^{-m})$ for odd $m>4$. For designed distance $28\cdot 2^j$, we have a probabilistic algorithm with success probability $\geq 1/3-O(2^{-m/2})$ for even $m\geq 6$. Finally, for designed distance $120\cdot 2^j$, we have a deterministic algorithm for $m\geq 8$ divisible by $4$. We also present a construction via Gold functions when $2i|m$. Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who proved that for extended binary BCH codes of designed distance $d(m,s,i)$, the minimum distance equals the designed distance. Their proof makes use of a non-constructive result of Berlekamp (Inform. Contrl., 1970), and a constructive ``down-conversion theorem'' that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive argument of Berlekamp by a low-complexity algorithm. In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT, 2012), who presented explicit minimum-weight words for designed distance $6$ (and hence also for designed distance $6\cdot 2^j$, by a well-known ``up-conversion theorem''), as we cover more cases of the minimum distance. However, the minimum-weight words we construct are not affine generators for designed distance $>6$

Lower bounds for constant query affine-invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well-suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally correctable code (LCC), where $\mathbb{K}$ is a finite field and $\Sigma$ is a finite alphabet, then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-1}))$. Also, we show that if $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally testable code (LTC), then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-2}))$. The dependence on $n$ in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS `13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto a small error in the Gowers norm

The complex Lorentzian Leech lattice and the bimonster

We find 26 reflections in the automorphism group of the the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.Comment: 24 pages, 3 figures, revised and proof corrected. Some small results added. to appear in the Journal of Algebr

On classifying Minkowskian sublattices

Let $\Lambda$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lambda'$ be a Minkowskian sublattice of $\Lambda$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lambda$. We extend the classification of possible $\Z/d\Z$-codes of the quotients $\Lambda/\Lambda'$ to dimension~$9$, where $d\Z$ is the annihilator of $\Lambda/\Lambda'$.Comment: 34 pages; incorporated referee comment