On optimal codes with w-identifiable parent property

Abstract

Let $\mathcal{C}$ be a q-ary code of length n and $X \subseteq \mathcal{C}$, then d is called a descendant of X if d _i ∈ {x _i : x ∈X} for all 1 ≤ i ≤ n. $\mathcal{C}$ is said to be a w-identifiable parent property code (w-IPP code for short) if whenever d is a descendant of w (or fewer) codewords, one can always identify at least one of the parent codewords in $\mathcal{C}$. In this paper, we give constructions for w-IPP codes of length w + 1. Furthermore, we show that F _w (w + 1,q), the maximum cardinality of a w-IPP q-ary code of length w + 1, satisfies $|\fancyscript{G}_{h(q)}| \leq F_{w}(w+1,q) \leq |\fancyscript{G}_{h(q)}|+6$, where $\fancyscript{G}_{h(q)}$ is a well-defined code graph. Finally, we give an efficient (O(q ^w+1)) algorithm to find the values of F _w (w + 1,q).