Complexity of Decoding Positive-Rate Reed-Solomon Codes

The complexity of maximal likelihood decoding of the Reed-Solomon codes $[q-1, k]_q$ is a well known open problem. The only known result in this direction states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, we remove the rate restriction and prove that the same complexity result holds for any positive information rate. In particular, this resolves an open problem left in [4], and rules out the possibility of a polynomial time algorithm for maximal likelihood decoding problem of Reed-Solomon codes of any rate under a well known cryptographical hardness assumption. As a side result, we give an explicit construction of Hamming balls of radius bounded away from the minimum distance, which contain exponentially many codewords for Reed-Solomon code of any positive rate less than one. The previous constructions only apply to Reed-Solomon codes of diminishing rates. We also give an explicit construction of Hamming balls of relative radius less than 1 which contain subexponentially many codewords for Reed-Solomon code of rate approaching one

On the hardness of learning sparse parities

This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EvenSet problem: given a homogeneous system of linear equations over F_2 on n variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EvenSet has been a notorious open problem. Towards this goal, we show that unless k-Clique can be solved in n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0. Our work also shows that the non-homogeneous generalization of the problem -- which we call k-VectorSum -- is W[1]-hard on instances where the number of equations is O(k log n), improving on previous reductions which produced Omega(n) equations. We also show that for any constant eps > 0, given a system of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or if every function on at most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the equations. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities. In a similar vein, we use the hardness of k-EvenSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time there is no poly(m, n)2^{o(sqrt{k}) time algorithm to decide whether a given set of m points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx. Pr_{F_2^n}[P(z) = 0] fraction of the points i.e., P is fooled by the set of points

Les codes algébriques principaux et leur décodage

National audienceLe premier exposé reprend les algorithmes classiques de décodage des codes géométriques, basés sur l'algorithme de Berlekamp-Massey et ses généralisations multivariées (Berlekamp-Massey-Sakata). Toutefois, avant de présenter ces algorithmes, je rappelerai les bases de la théorie des codes : codes linéaires, borne de Singleton, codes de Reed-Solomon, borne de Hamming. Ensuite, j'introduirai de manière motivée la famille des codes géométriques, comme généralisation des codes géométriques, après un bref rappel de la théorie des courbes algébriques sur les corps finis. La cadre sera alors en place pour introduire le décodage par syndrômes, qui est le décodage classique des codes géométriques. Le deuxième exposé est consacré aux progrès récents dans le domaine du codage algébrique, qui reposent sur le décodage par interpolation. Ces progrès sont dus à Guruswami-Sudan, et reposent sur une vision duale des codes de Reed-Solomon et des codes géométriques. Je présenterai dans l'ordre les algorithmes de Berlekamp-Welsh, Sudan et Guruswami-Sudan, dans le contexte des codes de Reed-Solomon et dans le contexte des codes géométriques. On verra finalement comment l'algorithme de Berlekamp-Massey-Sakata peut être recyclé dans ce contexte