On optimal codes with w-identifiable parent property
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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).
Keywords
w-identifiable parent property Graph theory Code graphAMS Classification
68R10 90C27 90C47Preview
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