The 4-GDDs of type 3562

We determine the 22 isomorphism classes of 4-GDDs with group type $3^5{6}^2$ and compare their automorphism groups. We also observe for the first time an example of a three dimensional Pasch triangle in a GDD.     

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Quasi-symmetric 2-(64,24,46) designs derived from AG(3,4)

This paper completes the enumeration of quasi-symmetric 2-$(64,24,46)$ designs supported by the dual code $C^{\perp }$ of the binary linear code $C$ spanned by the lines of $AG(3,4)$, initiated in Rodrigues and Tonchev (2015). It is shown that $C^{\perp }$ supports exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs. The automorphism groups of the related strongly regular graphs are computed.     

Structures of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra

In this paper, we determine the universal central extension, derivation algebra and automorphism group of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra. Furthermore, we generalize these results to the generalized twisted N = 1 Schrödinger–Neveu–Schwarz algebra in the final section.     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Automorphism groups of designs with λ=1

If $G$ is a finite group and $k=q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a 2-$(v,k,1)$-design $\mathbf{D}$ for which $\mathrm{Aut}\mathbf{D}?G$.