On the subset sum problem over finite fields

SIAM Journal on Computing, 2018

Establishing the complexity of Bounded Distance Decoding for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length N and dimension K = O(N), we show that it is NP-hard to decode more than N − K − c log N log log N errors (with c > 0 an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount > N − K − c log N (with c > 0 an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a Polynomial Reconstruction problem. In this view, our results show that it is NP-hard to decide whether there exists a degree K polynomial passing through K + c log N log log N points from a given set of points (a 1 , b 1), (a 2 , b 2). .. , (a N , b N). Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree K polynomial passing through K + c log N many points. These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called Moments Subset Sum, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.

Moscow Mathematical Journal

We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with coefficients in the finite field Fq with q elements, when d < q. It is shown that this formula holds in the affirmative for several values of r. In the general case, we give explicit lower and upper bounds for er(d, m) and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.

2007

We begin with the definition of Reed-Solomon codes. Definition 1.1 (Reed-Solomon code) . Let Fq be a finite field andFq[x] denote theFq-space of univariate polynomials where all the coefficients of x are fromFq. Pick {α1, α2, ...αn} distinct elements (also calledevaluation points ) of Fq and choosen and k such thatk ≤ n ≤ q. We define an encoding function for Reed-Solomon code as RS : Fq → F n q as follows. A message m = (m0, m1, ..., mk−1) with mi ∈ Fq is mapped to a degree k − 1 polynomial.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

IEEE Transactions on Information Theory, 2019

Let Fq be the finite field of q elements. In this paper we obtain bounds on the following counting problem: given a polynomial f (x) ∈ Fq[x] of degree k + m and a non-negative integer r, count the number of polynomials g(x) ∈ Fq[x] of degree at most k − 1 such that f (x) + g(x) has exactly r roots in Fq. Previously, explicit formulas were known only for the cases m = 0, 1, 2. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code [q, k, q−k+1]q.

Genevi eve Arboit Master In a 1996 paper, R. Impagliazzo and M. Naor show two average case reductions for the Subset Sum problem (SS). We use similar ideas to obtain stronger and additional such reductions for SS. Furthermore, we use modi cations of these ideas to obtain similar reductions for the Decoding of Linear Codes problem (DLC). The theorems give further evidence that the hardest case for Average case SS is when the number of integers is equal to their length. For Average case DLC, the theorems give evidence that the hardest case is when the dimension of the code is equal to the channel capacity times the length of the words.

Acta Arithmetica, 2002

arXiv (Cornell University), 2024

Let Fq be the finite field of q elements, for a given subset D ⊂ Fq, m ∈ N, an integer k ≤ |D| and b ∈ F m q we are interested in determining the existence of a subset S ⊂ D of cardinality k such that a∈S a i = bi for i = 1,. .. , m. This problem is known as the moment subset sum problem and it is N P-complete for a general D. We make a novel approach of this problem trough algebraic geometry tools analyzing the underlying variety and employing combinatorial techniques to estimate the number of Fq-rational points on certain varieties. We managed to give estimates on the number of Fq-rational points on certain diagonal equations and use this results to give estimations and existence results for the subset sum problem.

Finite Fields and Their Applications, 2012

Let F q be a finite field with q = p m elements, where p is an odd prime and m 1. In this paper, we explicitly determine all the μ-constacyclic codes of length 2 n over F q , when the order of μ is a power of 2. We further obtain all the self-dual negacyclic codes of length 2 n over F q and give some illustrative examples. All the repeated-root λ-constacyclic codes of length 2 n p s over F q are also determined for any nonzero λ in F q. As examples all the 2-constacyclic, 3-constacyclic codes of length 2 n 5 s over F 5 and all the 3-constacyclic, 5-constacyclic codes of length 2 n 7 s over F 7 for n 1, s 1 are derived.

Acta Arithmetica, 2010

Discrete Applied Mathematics, 2011

In this paper, we investigate some algebraic and combinatorial properties of a special Boolean function on n variables, defined using weighted sums in the residue ring modulo the least prime p ≥ n. We also give further evidence to a question raised by Shparlinski regarding this function, by computing accurately the Boolean sensitivity, thus settling the question for prime number values p = n. Finally, we propose a generalization of these functions, which we call laced functions, and compute the weight of one such, for every value of n.

European Journal of Pure and Applied Mathematics

A cyclic code has been one of the most active research topics in coding theory because they have many applications in data storage systems and communication systems. They have efficient encoding and decoding algorithms. This paper explains the construction of a family of cyclic codes from sequences generated by a trace of a monomial over finite fields of odd characteristics. The parameter and some examples of the codes are presented in this paper.

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

Journal of Discrete Mathematics, 2014

The problem of finding the number of irreducible monic polynomials of degreeroverFqnis considered in this paper. By considering the fact that an irreducible polynomial of degreeroverFqnhas a root in a subfieldFqsofFqnrif and only if(nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields ofFqnr. We also use the lattice of subfields ofFqnrto determine if it is possible to generate a Goppa code using an element lying in a proper subfield ofFqnr.

arXiv (Cornell University), 2019

In this paper we describe a class of codes called permutation codes. This class of codes is a generalization of cyclic codes and quasi-cyclic codes. We also give some examples of optimal permutation codes over binary, ternary, and 5-ary. Then, we describe its structure as submodules over a polynomial ring.