Related papers: Some complementary Gray codes
The automorphism group of a binary doubly-even self-dual code is always contained in the alternating group. On the other hand, given a permutation group $G$ of degree $n$ there exists a doubly-even self-dual $G$-invariant code if and only…
Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality…
It is well known that quantum codes can be constructed by means of classical symplectic dual-containing codes. This paper considers a family of two-generator quasi-cyclic codes and derives sufficient conditions for these codes to be…
A new method to construct $q$-ary complementary sequence (or array) sets (CSSs) and complete complementary codes (CCCs) of size $N$ is introduced in this paper. An algorithm on how to compute the explicit form of the functions in…
An important question in the study of quasi-perfect codes is whether such codes can be constructed for all possible lengths $n$. In this paper, we address this question for specific values of $n$. First, we investigate the existence of…
We consider the following combinatorial question. Let $$ S_0 \subset S_1 \subset S_2 \subset ...\subset S_m $$ be nested sets, where #$(S_i) = i$. A move consists of altering one of the sets $S_i$, $1 \le i \le m-1$, in a manner so that the…
Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes…
The results of this thesis allows one to replace calculations in tricategories with equivalent calculations in Gray categories (aka semistrict tricategories). In particular the rewriting calculus for Gray categories as used for example by…
This paper considers the construction of isodual quasi-cyclic codes. First we prove that two quasi-cyclic codes are permutation equivalent if and only if their constituent codes are equivalent. This gives conditions on the existence of…
An $n$-length binary word is $q$-decreasing, $q\geq 1$, if every of its length maximal factor of the form $0^a1^b$ satisfies $a=0$ or $q\cdot a > b$.We show constructively that these words are in bijection with binary words having no…
An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography,…
Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily…
The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the…
Complementarity is one of the main features of quantum physics that radically departs from classical notions. Here we consider the limitations that this principle imposes due to the unpredictability of measurement outcomes of incompatible…
Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle \gamma\rangle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle \gamma\rangle$. In this paper, the algebraic structure of cyclic…
We describe a generalization of the group testing problem termed symmetric group testing. Unlike in classical binary group testing, the roles played by the input symbols zero and one are "symmetric" while the outputs are drawn from a…
Non-binary codes correcting multiple deletions have recently attracted a lot of attention. In this work, we focus on multiplicity-free codes, a family of non-binary codes where all symbols are distinct. Our main contribution is a new…
Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes of odd dimension are shown to be consta-dihedral. Exact counting formulae are derived for DN codes. The special class of…
We show that any ternary Euclidean (resp.\ quaternary Hermitian) linear complementary dual $[n,k]$ code contains a Euclidean (resp.\ Hermitian) linear complementary dual $[n,k-1]$ subcode for $2 \le k \le n$. As a consequence, we derive a…
Linear complementary dual (LCD) codes introduced by Massey are the codes whose intersections with their dual codes are trivial. It can help to improve the security of the information processed by sensitive devices, especially against…