LCD codes constructed from weakly <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-self-orthogonal 1-designs

By a suitable extension of incidence matrices, orbit matrices, and submatrices of orbit matrices of weakly $ p $-self-orthogonal $ 1 $-designs, in this paper we describe a construction of LCD codes over a finite field. To illustrate the construction, we determine some binary LCD codes from self-orthogonal $ 1 $-designs which are invariant under the transitive non-regular action of the alternating group of order $ 60 $. With the described construction, we have obtained $ 6 $ optimal codes and $ 3 $ near-optimal codes.