Constructions of r-identifying codes and (r, ≤ l)-identifying codes

Abstract

In binary Hamming space, we construct r-identifying codes from a $$\sum\limits_{i = 0}^{r - 1} {\left({\matrix{n \cr i \cr}} \right) + \left({\matrix{{n - 1} \cr {r - 1} \cr}} \right) + 1} $$ -fold r-covering code. By using this construction, we modify the construction of r-identifying codes of Charon et al. which helps to find codes of greater length. From this modified construction, we improve the upper bound of M5 (23) which is the smallest possible cardinality of a 5-identifying code of length 23. We also construct (2, ≤ l) -identifying codes and we define the length function L(r, l) as the smallest positive integer n for which there exists an (r, ≤ l)-identifying code in $$\mathbb{F}_{2}^{n}$$ . By using the construction of (2, ≤ l)-identifying codes, we improve the upper bounds of L(2, l) for all l ≥ 6. We also improve the upper bounds of M 2 (≤ ) (n) for all l ≥ 6 and when n is one more than the improved upper bound of L(2, l). We give the construction for (r, ≤ l)-identifying codes. From this result, we prove that M (≤ ) (2r +1) ≤ 22r for certain values of r and l.