Special sequences as subcodes of reed-solomon codes

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 *

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 *

Designs, Codes and Cryptography, 2013

This paper addresses the question of how often the square code of an arbitrary l-dimensional subcode of the code GRS k (a, b) is exactly the code GRS 2k−1 (a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first stated and used in where Wieschebrink applied the Sidelnikov-Shestakov attack [8] to brake the Berger-Loidreau cryptostystem .

2021 IEEE Information Theory Workshop (ITW)

We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection metric. In sharp contrast with the behavior of linear block codes, we show that the typical non-linear code in the Hamming metric of cardinality q n−d+1 is far from having minimum distance d, i.e., from being MDS. We also give more precise results about the asymptotic proportion of block codes with good distance properties within the set of codes having a certain cardinality. We then establish the analogous results for subspace codes with the injection metric, showing also an application to the theory of partial spreads in finite geometry.

ArXiv, 2021

Abstract. For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes C1, C2, C3, . . . with parameters [ni, ki, di] such that kidi/ni grows quickly in the sense that kidi/ni &gt; √ ki − 1 &gt; 2i − 1. Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F2 and kidi/ni is asymptotic to 3n i / √

Siam Journal on Discrete Mathematics, 2010

A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F (n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove lim F (n, 3)/ n 3 = 5/4, more exactly, 5 24 n 3 − 1 2 n 2 − O(n log n) < F (n, 3) ≤ 5 24 n 3 − 1 2 n 2 + 5 8 n, and here equality holds when n is a power of 3. Many problems remain open.

Designs, Codes and Cryptography, 2007

We determine the minimum length n q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n q (k, d) = g q (k, d) + 1 for q k−1 − 2q k−1 2 −q + 1 ≤ d ≤ q k−1 − 2q k−1 2 when k is odd, for q k−1 − q k 2 − q k 2 −1 − q + 1 ≤ d ≤ q k−1 − q k 2 − q k 2 −1 when k is even, and for 2q k−1 − 2q k−2 − q 2 − q + 1 ≤ d ≤ 2q k−1 − 2q k−2 − q 2 .

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.

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.