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LCD codes obtained from weakly p-self-orthogonal designs

A 1-design is weakly p-self-orthogonal if all the block intersection numbers give the same residue modulo p. In [1], we analyze extensions of the incidence matrix, orbit matrix, and submatrices of the orbit matrix of a weakly p-self-orthogonal 1-design in order to construct self-orthogonal codes. A linear code is called LCD code if the intersection with its dual code is trivial. Matrix G generates an LCD code if and only if det(GG^T) is nonzero (see [3]). We extend the methods of construction described in [1] in order to construct LCD codes over finite fields. We use suitable extensions of incidence matrix, orbit matrices, and submatrices of orbit matrices in order to construct LCD codes over a finite field. We will present examples of LCD codes constructed from weakly p-self- orthogonal designs obtained from groups using the construction described in [2]. [1] V. Mikulić Crnković, I. Traunkar, Self- orthogonal codes constructed from weakly self- orthogonal designs invariant under an action of M_{; ; 11}; ; , AAECC (2021), https://doi.org/10.1007/s00200-020-00484-2. [2] D. Crnković, V. Mikulić Crnković, A. Švob, On some transitive combinatorial structures constructed from the unitary group U(3, 3), J. Statist. Plann. Inference, 144 (2014), 19-40. [3] J. L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992), 337–342.

weakly p-self orthogonal designs, LCD codes

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