New bounds on binary identifying codes

The original motivation for identifying codes comes from fault diagnosis in multiprocessor systems. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.In this paper, we concentrate on identification in binary Hamming spaces. We give a new lower bound on the cardinality of rr-identifying codes when r≥2r≥2. Moreover, by a computational method, we show that M1(6)=19M1(6)=19. It is also shown, using a non-constructive approach, that there exist asymptotically good (r,≤ℓ)(r,≤ℓ)-identifying codes for fixed ℓ≥2ℓ≥2. In order to construct (r,≤ℓ)(r,≤ℓ)-identifying codes, we prove that a direct sum of rr codes that are (1,≤ℓ)(1,≤ℓ)-identifying is an (r,≤ℓ)(r,≤ℓ)-identifying code for ℓ≥2ℓ≥2.