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 $\mathbb{Z}_4$‎, ‎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 $\mathbb{Z}_4$ 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 the other words‎,
‎\begin{equation*}‎
‎d_m^L(D)=min\lbrace wt_L(E); E \text{ is a } \mathbb{Z}_4-\text{submodule of D} \text{‎ , ‎rank(E)}=m\rbrace‎.
‎\end{equation*}‎
‎It is obtained that for $m‎, ‎1\leq m \leq rank(D),$We have‎
‎\begin{equation*}‎
‎\lfloor\dfrac{d_m^L(D)-2m+1}{2}\rfloor \leq l-rank(D)‎.
‎\end{equation*}‎
‎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
‎\begin{equation*}‎
‎d_m^H(D)=min\lbrace \vert supp(E) \vert ; E \text{ is a } \mathbb{Z}_4-\text{submodule of D} \text{ and rank(E)}=m\rbrace‎,
‎\end{equation*}‎
‎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 details‎.

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