356 research outputs found
Completely regular codes with different parameters giving the same distance-regular coset graphs
We construct several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a,b, we construct completely transitive codes over different fields with covering radius Ď=min{a,b}Ď=min{a,b} and identical intersection array, specifically, one code over F_q^r for each divisor r of a or b. As a corollary, for any prime power qq, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters
Families of nested completely regular codes and distance-regular graphs
In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius equal to or ,
and are -th parts, for of binary (respectively,
extended binary) Hamming codes of length (respectively, ), where
. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter equal to or are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive
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 strongly walk regular graphs,triple sum sets and their codes
Strongly walk-regular graphs (SWRGs) can be constructed as coset graphs of
the duals of projective three-weight codes whose weights satisfy a certain
equation. We provide classifications of the feasible parameters of these codes
in the binary and ternary case for medium size code lengths. For the binary
case, the divisibility of the weights of these codes is investigated and
several general results are shown.
It is known that an SWRG has at most 4 distinct eigenvalues . For an -SWRG, the triple satisfies a certain homogeneous polynomial equation of degree (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we
use methods from algorithmic arithmetic geometry to show that for and
, there are only the obvious solutions, and we conjecture this to remain
true for all (odd) .Comment: 42 page
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