The Multicovering Radii of Codes Preliminary Version
نویسنده
چکیده
The covering radius of a code is the least r such that the set of balls of radius r around codewords covers the entire ambient space. We introduce a generalization of the notion of covering radius. The m-covering radius of a code is the least radius such that the set of balls of the radius covers all m-tuples of elements in the ambient space. We investigate basic properties of m-covering radii. We investigate whether codes exist with given m-covering radii (they don’t always). We derive bounds on the size of the smallest code with a given m-covering radius, based on generalizations of the sphere bound and the method of counting excesses.
منابع مشابه
Multicovering Bounds from Relative Covering Radii
The multicovering radii of a code are recently introduced natural generalizations of the covering radius measuring the smallest radius of balls around codewords that cover all m-tuples of vectors. In this paper we prove a new identity relating the multicovering radii of a code to a relativized notion of ordinary covering radius. This identity is used to prove new bounds on the multicovering rad...
متن کاملBounds for the Multicovering Radii of Reed-Muller Codes with Applications to Stream Ciphers
The multicovering radii of a code are recent generalizations of the covering radius of a code. For positive m, the m-covering radius of C is the least radius t such that every m-tuple of vectors is contained in at least one ball of radius t centered at some codeword. In this paper upper bounds are found for the multicovering radii of first order Reed-Muller codes. These bounds generalize the we...
متن کاملThe multicovering radius problem for some types of discrete structures
The covering radius problem is a question in coding theory concerned with finding the minimum radius r such that, given a code that is a subset of an underlying metric space, balls of radius r over its code words cover the entire metric space. Klapper ([13]) introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduc...
متن کاملImproved lower bounds for multicovering codes
The m-covering radius of a code is a recent generalization of the covering radius of a code. It is the smallest t such that every m-tuple of vectors is contained in a ball of Hamming radius centered at some codeword. We derive new lower bounds for the size of the smallest code that has a given length and m-covering radius.
متن کاملOn the covering radii of extremal doubly even self-dual codes
In this note, we study the covering radii of extremal doubly even self-dual codes. We give slightly improved lower bounds on the covering radii of extremal doubly even self-dual codes of lengths 64, 80 and 96. The covering radii of some known extremal doubly even self-dual [64, 32, 12] codes are determined.
متن کامل