Bounds and constructions for $${\overline{3}}$$ 3 ¯ -separable codes with length 3

Abstract

Abstract. Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let C be a code of length n over an alphabet of q letters. The descendant code desc(C0) of C0 = {c1, c2, . . . , ct} ⊆ C is defined to be the set of words x = (x1, x2, . . . , xn)T such that xi ∈ {c1,i, c2,i, . . . , ct,i} for all i = 1, . . . , n, where cj = (cj,1, cj,2, . . . , cj,n)T . C is a t-separable code if for any two distinct C1, C2 ⊆ C with |C1| ≤ t, |C2| ≤ t, we always have desc(C1) = desc(C2). Let M (t, n, q) denote the maximal possible size of such a separable code. In this paper, an upper bound on M (3, 3, q) is derived by considering an optimization problem related to a partial Latin square, and then two constructions for 3-SC(3, M, q)s are provided by means of perfect hash families and Steiner triple systems.

Bounds and constructions for 3 ¯ $\overline {3}$ -strongly separable codes with length 3

“…To our best knowledge, the lower bound in [7] is the best known result on 3-SC(3, M, q). In the following section, we will improve the lower bound in Theorem 2.21 to Ω(q 5/3 + q 4/3 − q) for some prime powers q.…”

Bounds and Constructions for $\overline{3}$-Strongly Separable Codes with Length $3$

“…As an important class of anti-collusion codes in multimedia copyright protection, separable codes and frameproof codes were widely studied, e.g., [3,5,6,7,8,9,11].…”