On MLDR and MHDR codes

Farhang Baftani, Farzaneh

doi:10.22061/jdma.2025.12537.1167

On MLDR and MHDR codes

Document Type : Full Length Article

Author

Farzaneh Farhang Baftani

Department of Mathematics, Ard.C, Islamic Azad University

10.22061/jdma.2025.12537.1167

Abstract

For a code D of length l over ℤ₄, we denote by M(D) the matrix containing all code words of D on its rows. Any columns of M(D) corresponds to the column which is zero or it has zero and 2 equally or it has all elements of ℤ₄ equally. The Lee Weight for these columns is defined 0, 2 and 1, respectively. If we calculate the sum of all Lee weights of columns of M(D), it is denoted by wt_L(D) and called the Lee Support Weight of D. In addition, the m-th Generalized Lee Weight (GLW) for D, denoted by d_m^L(D), is defined as the minimum of the Lee Support Weights of all submodules of D of rank m. In other words, d_m^L(D) = min{wt_L(E) ; E is a ℤ₄-submodule of D, rank(E) = m}. It is obtained that for m, 1 ≤ m ≤ rank(D), we have ⌊(d_m^L(D) - 2m + 1) / 2⌋ ≤ l - rank(D). The code which meets the recent upper bound is called Maximum Lee Distance separable with respect to Rank (m-th MLDR) code. Also, if d_m^H(D) denotes the m-th GHW for code D, it is defined as d_m^H(D) = min{|supp(E)| ; E is a ℤ₄-submodule of D and rank(E) = m}. The upper bound for d_m^H(D) is l - rank(D). The code meeting this upper bound is called MHDR code. In this paper, we investigate MLDR codes, MHDR codes and relation between them, in detail.

Graphical Abstract