2-(56,12,3) designs and their class graphs

There exist exactly 1122 pairwise non-isomorphic 2-(56,12,3) designs being the residual designs of the known symmetric (71,15,3) designs. Six pairwise non-isomorphic strongly regular graphs with parameters (35,16,6,8) were constructed as class graphs of the obtained residual designs. Orders of the full automorphism groups of these graphs are 12, 32, 96, 192, 288 and 40320

New extremal Type II Z4\mathbb{Z}_4-codes of length 64 by the doubling method

Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$ The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$. Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before. Moreover, the residue codes of the constructed extremal $\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes form self-orthogonal $1$-designs

On some new extremal Type II Z4-codes of length 40

Using the building-up method and a modification of the doubling method we construct new extremal Type II Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $k_1in {8,9,10,11,12,14,15}$, are the first examples of extremal Type II Z4-codes of given type and length 40 whose residue codes have minimum weight greater than or equal to 8. Further, we use minimum weight codewords for a construction of 1-designs, some of which are self-orthogonal

Some symmetric (47,23,11) designs

Up to isomorphism there are precisely fifty-four symmetric designs with parameters (47,23,11) admitting a faithful action of a Frobenius group of order 55. From these fifty-four designs one can construct 179 pairwise nonisomorphic 2-(23,11,10) designs as derived and 191 pairwise nonisomorphic 2-(24,12,11) designs as residual designs. We have determined full automorphism groups of all constructed designs. One of 2-(24,12,11) designs has full automorphism group of order 15840, isomorphic to the group M11 × 2, acting transitively on the set of points

New regular two-graphs on 38 and 42 vertices

All regular two-graphs having up to 36 vertices are known, and the first open case is the enumeration of two-graphs on 38 vertices. It is known that there are at least 191 regular two-graphs on 38 vertices and at least 18 regular two-graphs on 42 vertices. The number of descendants of these two-graphs is 6760 and 120, respectively. In this paper, we classify strongly regular graphs with parameters (41,20,9,10) having nontrivial automorphisms and show that there are exactly 7152 such graphs. We enumerate all regular two-graphs on 38 and 42 vertices with at least one descendant whose full automorphism group is nontrivial and establish that there are at least 194 regular two- graphs on 38 vertices and at least 752 regular two-graphs on 42 vertices. Furthermore, we construct descendants with trivial automorphism group of newly constructed two-graphs and increase the number of known strongly regular graphs with parameters (37,18,8,9) and (41,20,9,10) to 6802 and 18439 respectively. This significantly increases the number of known strongly regular graphs with parameters (41,20,9,10)

New examples of self-dual near-extremal ternary codes of length 48 derived from 2-(47,23,11) designs

In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for $145$ distinct values of the number $A_{12}$ of codewords of minimum weight 12, and raised the question about the existence of codes for other values of $A_{12}$. In this note, we use symmetric 2-$(47,23,11)$ designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for $150$ new values of $A_{12}$.Comment: 7 page

2-(56,12,3) designs and their class graphs

There exist exactly 1122 pairwise non-isomorphic 2-(56,12,3) designs being the residual designs of the known symmetric (71,15,3) designs. Six pairwise non-isomorphic strongly regular graphs with parameters (35,16,6,8) were constructed as class graphs of the obtained residual designs. Orders of the full automorphism groups of these graphs are 12, 32, 96, 192, 288 and 40320