Shahid Rajaee Teacher Training University Journal of Discrete Mathematics and Its Applications 2981-0809 11 2 2026 06 01 On MLDR and MHDR codes 123 129 12566 10.22061/jdma.2025.12537.1167 EN Farzaneh Farhang Baftani Department of Mathematics, Ard.C, Islamic Azad University Journal Article 2025 09 21 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 wtL(D) and called the Lee Support Weight of D. In addition, the m-th Generalized Lee Weight (GLW) for D, denoted by dmL(D), is defined as the minimum of the Lee Support Weights of all submodules of D of rank m. In other words, dmL(D) = min{wtL(E) ; E is a ℤ4-submodule of D, rank(E) = m}. It is obtained that for m, 1 ≤ m ≤ rank(D), we have ⌊(dmL(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 dmH(D) denotes the m-th GHW for code D, it is defined as dmH(D) = min{|supp(E)| ; E is a ℤ4-submodule of D and rank(E) = m}. The upper bound for dmH(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.

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