Distance Distribution in Reed-Solomon Codes

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.

2004

For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k] q , a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n−g, which contains at least n g /q g−k many codewords. Letĝ(n, k, q) be the smallest positive integer g such that n g /q g−k ≤ 1. One knows that *

2018

In this paper, we obtain an asymptotic formula for the number of codewords with a fixed distance to a given received word of degree $k+m$ in the standard Reed-Solomon code $[q, k, q-k+1]_q$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$.

Siam Journal on Computing, 2007

For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k] q , a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n−g, which contains at least n g /q g−k many codewords. Letĝ(n, k, q) be the smallest positive integer g such that n g /q g−k ≤ 1. One knows that *

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.

Applicable Algebra in Engineering, Communication and Computing, 2016

For any finite field F q with q elements, we study the set F (q,m) of functions from F m q into F q. We introduce a transformation that allows us to determine a linear system of q m+1 equations and q m+1 unknowns, which has for solution the Hamming distances of a function in F (q,m) to all the affine functions.

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.

Linear Algebra and its Applications, 2020

Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7, 10, 24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form ax + b 0 x q s + b 1 x q s+n + b 2 x q s+2n +. .. + b t−1 x q s+n(t−1) ∈ F q nt [x], with gcd(s, n) = 1. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a q n-polynomial. Finally, as an applications of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in PG(1, q 6) presented in [9].

2015 IEEE International Symposium on Information Theory (ISIT), 2015

Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005) show that given a Reed-Solomon code over a finite field F, of length n and dimension t, and given a target vector v ∈ F n , it is NP-hard to decide if there is a codeword that disagrees with v on at most n − t − 1 coordinates. Understanding the complexity of this Bounded Distance Decoding problem as the amount of error in the target decreases is an important open problem in the study of Reed-Solomon codes. In this work we generalize this result by proving that it is NP-hard to decide the existence of a codeword that disagrees with v on n − t − 2 and on n − t − 3 coordinates. No other NP-hardness results were known before for an amount of error < n − t − 1. The core of our proof is showing the NP-hardness of a parameterized generalization of the Subset-Sum problem to higher degrees (called Moments Subset-Sum) that may be of independent interest.

Lecture Notes in Computer Science, 2013

For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed-Solomon codes is proved to be co-NP-complete [9][5]. For the extended Reed-Solomon codes RSq(Fq, k), a conjecture was made to classify deep holes in [5]. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes RSp(D, k), where p is a prime, |D| > k p−1 2. Our techniques are built on the idea of deep hole trees, and several results concerning the Erdös-Heilbronn conjecture.