New bounds on $$\bar{2}$$ 2 ¯ -separable codes of length 2

Abstract

Let C be a code of length n over an alphabet of q letters. The descendant code desc(C 0 ) of C 0 = {c 1 , c 2 , . . . , c t } ⊆ C is defined to be the set of wordsThe study of separable codes is motivated by questions about multimedia fingerprinting for protecting copyrighted multimedia data. Let M(t, n, q) be the maximal possible size of such a separable code. In this paper, we provide an improved upper bound for M(2, 2, q) by a graph theoretical approach, and a new lower bound for M(2, 2, q) by deleting suitable points and lines from a projective plane, which coincides with the improved upper bound in some places. This corresponds to the bounds of maximum size of bipartite graphs with girth 6 and a construction of such maximal bipartite graphs.

Strongly separable codes

“…Lemma IV.5. ( [4], [5]) Let k ≥ 2 be a prime power. Then there is an optimal 2-SC(2, M, q) for any q ∈ {k 2 − 1,…”

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

“…This stimulates the investigation of separable codes with short length n = 2, 3. When n = 2, 3, upper bounds on M (2, n, q) were derived, and several infinite series of optimal 2-SC(n, M, q)s were constructed in [6,7]. When t ≥ 3, the structure of t-SCs becomes more complex so that little is known about t-SCs for t ≥ 3.…”

Asymptotically optimal 2 ¯ $\overline {2}$ -separable codes with length 4

“…Furthermore, some optimal 2-SCs of length 2 were obtained. Theorem 1 [4] For any positive integer q, M (2, 2, q 3 2 , and…”

“…For more details on separable codes, the interested reader may refer to [2,[4][5][6][7]. Consider the case (t, n) = (2, 4).…”