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AMC

In this paper we present a method for constructing
self-orthogonal codes from orbit matrices of $2$-designs
that admit an automorphism group $G$ which acts with
orbit lengths $1$ and $w$, where $w$ divides $|G|$.
This is a generalization of an earlier method proposed
by Tonchev for constructing self-orthogonal codes from
orbit matrices of $2$-designs with a fixed-point-free
automorphism of prime order. As an illustration of our
method we provide a classification of self-orthogonal
codes obtained from the non-fixed parts of the orbit
matrices of the symmetric $2$-$(56,11,2)$ designs,
some symmetric designs $2$-$(71,15,3)$ (and their
residual designs), and some non-symmetric $2$-designs,
namely those with parameters $2$-$(15,3,1)$,
$2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$,
respectively with automorphisms of order $p$,
where $p$ is an odd prime. We establish that
the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$
are optimal two-weight codes. Further, we construct
an optimal binary self-orthogonal $[16,5,8]$ code
from the non-fixed part of the orbit matrix of the
$2$-$(64,8,1)$ design with respect to an automorphism
group of order four.

AMC

The alternating group $A_8$, acts as a primitive rank-3 group of degree $35$ on the set of lines of $V_4(2)$ with line stabilizer isomorphic to $2^4:(S_3 \times S_3)$ and orbits of lengths 1, 16 and 18 respectively. This action defines the unique strongly regular $(35, 16, 6, 8)$ graph. The paper examines the binary (resp. ternary) codes spanned by the rows of this graph, and its complement. We establish some properties of the codes and use the geometry of the designs and graphs to give an account on the nature of some classes of codewords, in particular those of minimum weight. Further, we show that the codes with parameters $[35, 28, 4]_2,[35, 6, 16]_2,[35, 29, 3]_2,[28, 7, 12]_2,[28, 21,4]_2, [36, 7, 16]_2, [36,29,4]_2$ and $[64, 56, 4]_2$ are all optimal. In addition, we show that the codes with parameters $[35, 13, 12]_3, [35, 22, 5]_3,[35, 14, 11]_3, [35, 21, 6]_3$ are all near-optimal for the given length and dimension.

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