New nonbinary code bounds based on divisibility arguments

Sven C Polak

doi:10.1007/s10623-017-0366-0

Abstract

For q,n,d in mathbb {N}, let A_q(n,d) be the maximum size of a code C subseteq [q]^n with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds A_5(8,6) le 65, A_4(11,8)le 60 and A_3(16,11) le 29. These in turn imply the new upper bounds A_5(9,6) le 325, A_5(10,6) le 1625, A_5(11,6) le 8125 and A_4(12,8) le 240. Furthermore, we prove that for mu ,q in mathbb {N}, there is a 1–1-correspondence between symmetric (mu ,q)-nets (which are certain designs) and codes C subseteq [q]^{mu q} of size mu q^2 with minimum distance at least mu q - mu . We derive the new upper bounds A_4(9,6) le 120 and A_4(10,6) le 480 from these ‘symmetric net’ codes.

Highlights

For any m ∈ N, we write [m] := {1, . . . , m}
We prove that for μ, q ∈ N, there is a 1–1-correspondence between symmetric (μ, q)-nets and codes C ⊆ [q]μq of size μq2 with minimum distance at least μq − μ
In this paper we find new upper bounds on Aq (n, d), based on a divisibility-argument

Summary

Introduction

For any m ∈ N, we write [m] := {1, . . . , m}. Fix n, q ∈ N. In this paper we find new upper bounds on Aq (n, d) (for some q, n, d), based on a divisibility-argument. In some cases, it will sharpen a combination of the following two well-known upper bounds on Aq (n, d). The second bound follows from the observation that in a (n, d)q -code any symbol can occur at most Aq (n − 1, d) times at the first position. We will use the following observations (which are well known and often used in coding theory and combinatorics). I.e., the number of pairs of distinct words {u, v} ⊆ C with distance unequal to d is at most the leftmost term minus the rightmost term in (4). We derive some new upper bounds from these ‘symmetric net’ codes.

The divisibility argument

Codes from symmetric nets