2 research outputs found
Properties of Codes in the Johnson Scheme
Codes which attain the sphere packing bound are called perfect codes. The
most important metrics in coding theory on which perfect codes are defined are
the Hamming metric and the Johnson metric. While for the Hamming metric all
perfect codes over finite fields are known, in the Johnson metric it was
conjectured by Delsarte in 1970's that there are no nontrivial perfect codes.
The general nonexistence proof still remains the open problem. In this work we
examine constant weight codes as well as doubly constant weight codes, and
reduce the range of parameters in which perfect codes may exist in both cases.
We start with the constant weight codes. We introduce an improvement of Roos'
bound for one-perfect codes, and present some new divisibility conditions,
which are based on the connection between perfect codes in Johnson graph J(n,w)
and block designs. Next, we consider binomial moments for perfect codes. We
show which parameters can be excluded for one-perfect codes. We examine
two-perfect codes in J(2w,w) and present necessary conditions for existence of
such codes. We prove that there are no two-perfect codes in J(2w,w) with length
less then 2.5*10^{15}. Next we examine perfect doubly constant weight codes. We
present a family of parameters for codes whose size of sphere divides the size
of whole space. We then prove a bound on length of such codes, similarly to
Roos' bound for perfect codes in Johnson graph. Finally we describe Steiner
systems and doubly Steiner systems, which are strongly connected with the
constant weight and doubly constant weight codes respectively. We provide an
anticode-based proof of a bound on length of Steiner system, prove that doubly
Steiner system is a diameter perfect code and present a bound on length of
doubly Steiner system.Comment: This is an M.Sc.thesis submitted in February 2007 by Natalia
Silberstein and supervised by Prof. Tuvi Etzio
On Perfect Codes in the Johnson Graph
In this paper we consider the existence of nontrivial perfect codes in the
Johnson graph J(n,w). We present combinatorial and number theory techniques to
provide necessary conditions for existence of such codes and reduce the range
of parameters in which 1-perfect and 2-perfect codes may exist.Comment: Submitted for ACCT 201