Galois LCD codes over mixed alphabets

Some constructions of $l$-Galois LCD codes

“…They provided necessary and sufficient conditions for generalized Galois LCD codes over chain rings. Recently, in 2023, Bajalan et al [2] characterized Galois LCD codes and Galois-invariant codes over mixed alphabets of finite chain rings. There are some other works in which Galois inner product was taken into consideration to study linear codes [6], and their applications in quantum codes were also presented (see [14,15]).…”

“…and det(G[σ(G)] t ) = 2 = 0. Therefore, C is a 0-Galois LCD code with the parameters [6,3,2] 3 which is an almost optimal code, i.e., the minimum distance is one less than that of the optimal code (according to Grassl table [8] available online). where I is the identity matrix of order 2 and where I is the identity matrix of order 4 and…”

“…Therefore, C is a 1-Galois LCD code with the parameters [8,4,2] 4 . Now, if we eliminate the first three zero columns from the above matrix G, then the corresponding code will be a 1-Galois LCD [5,4,2] 4 -code, which is an MDS code.…”

“…satisfies the condition det(G[σ(G)] t ) = 2 = 0. Therefore, the linear code generated by the matrix G is a 0-Galois LCD code with the parameters [8,4,2] 3 . Now, suppose we eliminate the zero column from the above matrix G. In that case, the corresponding code will be a 0-Galois LCD [7,4,2] 3 -code, which is an almost optimal code, i.e., the minimum distance is one less than that of the optimal code (according to Grassl table [8] available online).…”

“…Therefore, the linear code generated by the matrix G is a 0-Galois LCD code with the parameters [8,4,2] 3 . Now, suppose we eliminate the zero column from the above matrix G. In that case, the corresponding code will be a 0-Galois LCD [7,4,2] 3 -code, which is an almost optimal code, i.e., the minimum distance is one less than that of the optimal code (according to Grassl table [8] available online).…”