On optimal codes with w-identifiable parent property

Let C be a q-ary code of length n and X ⊆ C, then d is called a descendant of X if di ∈ {xi : x ∈ X} for all 1 ≤ i ≤ n. 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 C. In this paper, we give constructions for w-IPP codes of length w + 1. Furthermore, we show that Fw(w + 1, q), the maximum cardinality of a w-IPP q-ary code of length w + 1, satisfies |Gh(q)| ≤ Fw(w + 1, q) ≤ |Gh(q)|+6, where Gh(q) is a well-defined code graph. Finally, we give an efficient (O(qw+1)) algorithm to find the values of Fw(w + 1, q).