Nonexistence Theorems for Perfect Codes over Finite Fields

In this survey, we consider the trivial perfect codes which are the binary repetition codes of odd length and all codes that only contain one code word. We also take into acount, the Hamming codes as well as the two perfect Golay codes. We then prove that there do not exist any perfect codes over finite fields others than the ones above, using Lloyd's theorem

Two theorems on perfect codes

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On t -designs

Introduction and preliminaries An incidence structure is a triple S=(X, <J,9 S} where X and JL are disjoint sets andc^ci^fX^?. Elements x^X are called points and elements A^JL are called blocks of S. A point x and a block ^4 are incident iff (#, ^4)ec?. For any block A, (A) will denote the set of points incident with A

Weighted Coverings and Packings

In this paper we introduce a generalization of the concepts of coverings and packings in Hamming space called weighted coverings and packings. This allows us to formulate a number of well-known coding theoretical problems in a uniform manner. We study the existence of perfect weighted codes, discuss connections between weighted coverings and packings, and present many constructions for them