arXiv subjects
Publications and source records attributed to Nadja Willenborg.
We establish dihedral quantum codes of short block length, a class of CSS codes obtained by the lifted product construction. We present the code construction and give a formula for the code dimension, depending on the two classical codes that the CSS code is based on. We also give a lower bound on the code distance and construct an example of short dihedral quantum codes.
Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $λ\in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes.
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.
We consider $t$-Lee-error-correcting codes of length $n$ over the residue ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower bounds on the number of $t$-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.
Let $w_m(G)$ denote the total number of walks of length $m$ in an undirected graph $G$. Spectral decomposition shows that $(w_m(G))_{m\ge 0}$ is a moment sequence of a finite positive measure. We use exchangeability of its product measures to turn global nonnegativity of polynomial symmetrizations into universal inequalities for the number of walks. For exponent vectors $\alpha,\beta$ in distinct permutation orbits, $\mathrm{Sym}(x^\beta-x^\alpha)$ is globally nonnegative exactly when $\beta$ is coordinatewise even and majorizes $\alpha$. This finite criterion includes several classical inequalities as special cases; univariate polynomials and squared alternants also yield linear and Hankel determinant inequalities.
We determine the asymptotic proportion of free modules over finite chain rings with good distance properties and treat the asymptotics in the code length n and the residue field size q separately. We then specialize and apply our technique to rank metric codes and to Hamming metric codes.