Improved bounds on identifying codes in binary Hamming spaces

Let ℓ , n and r be positive integers. Define F n = { 0 , 1 } n . The Hamming distance between words x and y of F n is denoted by d ( x , y ) . The ball of radius r is defined as B r ( X ) = { y ∈ F n ∣ ∃ x ∈ X : d ( x , y ) ≤ r } , where X is a subset of F n . A code C ⊆ F n is called ( r , ≤ ℓ ) -identifying if for all X , Y ⊆ F n such that | X | ≤ ℓ , | Y | ≤ ℓ and X ≠ Y , the sets B r ( X ) ∩ C and B r ( Y ) ∩ C are different. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. s paper, we present various results concerning ( r , ≤ ℓ ) -identifying codes in the Hamming space F n . First we concentrate on improving the lower bounds on ( r , ≤ 1 ) -identifying codes for r > 1 . Then we proceed by introducing new lower bounds on ( r , ≤ ℓ ) -identifying codes with ℓ ≥ 2 . We also prove that ( r , ≤ ℓ ) -identifying codes can be constructed from known ones using a suitable direct sum when ℓ ≥ 2 . Constructions for ( r , ≤ 2 ) -identifying codes with the best known cardinalities are also given.