46,003 research outputs found
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
On the binary codes with parameters of triply-shortened 1-perfect codes
We study properties of binary codes with parameters close to the parameters
of 1-perfect codes. An arbitrary binary code ,
i.e., a code with parameters of a triply-shortened extended Hamming code, is a
cell of an equitable partition of the -cube into six cells. An arbitrary
binary code , i.e., a code with parameters of a
triply-shortened Hamming code, is a cell of an equitable family (but not a
partition) from six cells. As a corollary, the codes and are completely
semiregular; i.e., the weight distribution of such a code depends only on the
minimal and maximal codeword weights and the code parameters. Moreover, if
is self-complementary, then it is completely regular. As an intermediate
result, we prove, in terms of distance distributions, a general criterion for a
partition of the vertices of a graph (from rather general class of graphs,
including the distance-regular graphs) to be equitable. Keywords: 1-perfect
code; triply-shortened 1-perfect code; equitable partition; perfect coloring;
weight distribution; distance distributionComment: 12 page
- …