Partial sums of binomials, intersecting numbers, and the excess bound in Rosenbloom–Tsfasman space

Abstract

In this work, the sphere-covering bound on covering codes in Rosenbloom-Tsfasman spaces (RT spaces) is improved by generalizing the excess counting method. The approach focuses on studying the parity of a Rosenbloom-Tsfasman sphere (RT sphere) and the parity of the intersection of two RT spheres. We connect the parity of an RT sphere with partial sums of binomial coefficients and p-adic valuation of binomial coefficients. The intersection number of RT spaces is introduced and we determinate its parity under some conditions. Numerical applications of the method are discussed. Keywords Rosenbloom-Tsfasman metric • Covering code • Bounds on code • Sum of binomials • Congruence Mathematics Subject Classification 94B25 • 05A10 • 05A19 • 06A06 Communicated by Masaaki Harada.

Ordered covering arrays and upper bounds on covering codes

“…for a general NRT poset remains sparse [5][6][7]. One upper bound which we will improve is the general upper bound given in the next result.…”

“…We refer the reader to [5] for more details on covering codes in NRT spaces. In [5][6][7] the notation K m s R ( , , )…”

“…Central concepts about codes in Hamming spaces have been also investigated in NRT spaces, such as perfect codes, maximum distance separable (MDS) codes, linear codes, weight distribution, packing, and covering problems [3,5,6,24,26,29,31]. See the book by Firer et al [16] for further concepts and applications of the NRT metric to coding theory.…”

“…The general problem of covering codes in the NRT space is proposed by the first two authors in [5], which deals mainly with upper bounds, recursive relations, and some sharp bounds as well as relations with MDS codes. More recently, the sphere covering bound in NRT spaces is improved in [6] under some conditions by generalizing the excess counting method. In the present work, we explore upper bounds and recursive relations for covering codes in NRT spaces using ordered covering arrays (OCAs).…”

On the non-existence of perfect codes in the NRT-metric

“…On the one hand, the NRT metric is a special case of poset metric [1] (corresponding to the case when the poset is a disjoint union of chains of the same length) and on the other hand the NRT metric generalizes the Hamming metric (the latter corresponds to the special case when the poset is an antichain). Some central concepts on codes in Hamming spaces have been investigated in NRT spaces, such as perfect codes, MDS codes, weight distribution, packing and covering problems, see for instance [1,2,3,5,7,9]. This paper deals with the existence of perfect codes in NRT spaces.…”