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

Research paper by **Minquan Cheng, Jing Jiang, Haiyan Li, Ying Miao, Xiaohu Tang**

Indexed on: **03 Jul '15**Published on: **03 Jul '15**Published in: **Computer Science - Information Theory**

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#### Abstract

Separable codes were introduced to provide protection against illegal
redistribution of copyrighted multimedia material. Let $\mathcal{C}$ be a code
of length $n$ over an alphabet of $q$ letters. The descendant code ${\sf
desc}(\mathcal{C}_0)$ of $\mathcal{C}_0 = \{{\bf c}_1, {\bf c}_2, \ldots, {\bf
c}_t\} \subseteq {\mathcal{C}}$ is defined to be the set of words ${\bf x} =
(x_1, x_2, \ldots,x_n)^T$ such that $x_i \in \{c_{1,i}, c_{2,i}, \ldots,
c_{t,i}\}$ for all $i=1, \ldots, n$, where ${\bf
c}_j=(c_{j,1},c_{j,2},\ldots,c_{j,n})^T$. $\mathcal{C}$ is a
$\overline{t}$-separable code if for any two distinct $\mathcal{C}_1,
\mathcal{C}_2 \subseteq \mathcal{C}$ with $|\mathcal{C}_1| \le t$,
$|\mathcal{C}_2| \le t$, we always have ${\sf desc}(\mathcal{C}_1) \neq {\sf
desc}(\mathcal{C}_2)$. Let $M(\overline{t},n,q)$ denote the maximal possible
size of such a separable code. In this paper, an upper bound on
$M(\overline{3},3,q)$ is derived by considering an optimization problem related
to a partial Latin square, and then two constructions for
$\overline{3}$-SC$(3,M,q)$s are provided by means of perfect hash families and
Steiner triple systems.