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).