5 research outputs found
Permutation Inequalities for Walks in Graphs
Using spectral graph theory, we show how to obtain inequalities for the
number of walks in graphs from nonnegative polynomials and present a new family
of such inequalities
Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces
We investigate the asymptotic density of error-correcting codes with good
distance properties and prescribed linearity degree, including sublinear and
nonlinear codes. We focus on the general setting of finite
translation-invariant metric spaces, and then specialize our results to the
Hamming metric, to the rank metric, and to the sum-rank metric. Our results
show that the asymptotic density of codes heavily depends on the imposed
linearity degree and the chosen metric
On the Number of -Lee-Error-Correcting Codes
We consider -Lee-error-correcting codes of length over the residue
ring and determine upper and lower
bounds on the number of -Lee-error-correcting codes. We use two different
methods, namely estimating isolated nodes on bipartite graphs and the graph
container method. The former gives density results for codes of fixed size and
the latter for any size. This confirms some recent density results for linear
Lee metric codes and provides new density results for nonlinear codes. To apply
a variant of the graph container algorithm we also investigate some geometrical
properties of the balls in the Lee metric
Dihedral Quantum Codes
We study dihedral quantum codes of short block length, a large class of
quantum CSS codes obtained by the lifted product construction. We present code
construction and give a formula for the code dimension, depending on the two
classical codes on which the CSS code is based on. We also give a lower bound
on the code distance. Finally we construct an example of short dihedral quantum
codes, improving parameters of previously known quantum codes