On LCD codes from skew symmetric Toeplitz matrices.

A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let I be an identity matrix and T be a Toeplitz matrix of the same order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form (I , T). In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when T is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for T being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary. [ABSTRACT FROM AUTHOR]