Links Between Discriminating and Identifying Codes in the Binary Hamming Space
نویسندگان
چکیده
Let Fn be the binary n-cube, or binary Hamming space of dimension n, endowed with the Hamming distance, and En (respectively, On) the set of vectors with even (respectively, odd) weight. For r 1 and x 2 Fn, we denote by Br(x) the ball of radius r and centre x. A code C Fn is said to be r-identifying if the sets Br(x)\C, x 2 Fn, are all nonempty and distinct. A code C En is said to be r-discriminating if the sets Br(x) \ C, x 2 On, are all nonempty and distinct. We show that the two de nitions, which were given for general graphs, are equivalent in the case of the Hamming space, in the following sense: for any odd r, there is a bijection between the set of r-identifying codes in Fn and the set of r-discriminating codes in F n+1. We then extend previous studies on constructive upper bounds for the minimum cardinalities of identifying codes in the Hamming space.
منابع مشابه
Discriminating and Identifying Codes in the Binary Hamming Space
Let Fn be the binary n-cube, or binary Hamming space of dimension n, endowed with the Hamming distance, and En (respectively, On) the set of vectors with even (respectively, odd) weight. For r ≥ 1 and x ∈ Fn, we denote by Br(x) the ball of radius r and centre x. A code C ⊆ Fn is said to be r-identifying if the sets Br(x)∩C, x ∈ F n, are all nonempty and distinct. A code C ⊆ En is said to be r-d...
متن کاملConstructions for identifying codes
A nonempty set of words in a binary Hamming space F is called an r-identifying code if for every word the set of codewords within distance r from it is unique and nonempty. The smallest possible cardinality of an r-identifying code is denoted by Mr(n). In this paper, we consider questions closely related to the open problem whether Mt+r(n + m) ≤ Mt(m)Mr(n) is true. For example, we show results ...
متن کاملComputation of Minimum Hamming Weight for Linear Codes
In this paper, we consider the minimum Hamming weight for linear codes over special finite quasi-Frobenius rings. Furthermore, we obtain minimal free $R$-submodules of a finite quasi-Frobenius ring $R$ which contain a linear code and derive the relation between their minimum Hamming weights. Finally, we suggest an algorithm that computes this weight using the Grobner basis and we show that und...
متن کاملConnecting Yule Process, Bisection and Binary Search Tree via Martingales
We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process.
متن کاملLearning Cross-View Binary Identities for Fast Person Re-Identification
In this paper, we propose to learn cross-view binary identities (CBI) for fast person re-identification. To achieve this, two sets of discriminative hash functions for two different views are learned by simultaneously minimising their distance in the Hamming space, and maximising the cross-covariance and margin. Thus, similar binary codes can be found for images of a same person captured at dif...
متن کامل