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On MLDR and MHDR codes | ||
| Journal of Discrete Mathematics and Its Applications | ||
| دوره 11، شماره 2، شهریور 2026، صفحه 123-129 اصل مقاله (320.91 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2025.12537.1167 | ||
| نویسنده | ||
| Farzaneh Farhang Baftani* | ||
| Department of Mathematics, Ard.C, Islamic Azad University | ||
| تاریخ دریافت: 30 شهریور 1404، تاریخ بازنگری: 14 آبان 1404، تاریخ پذیرش: 01 دی 1404 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Lee Weight؛ Hamming weight؛ MHDR code؛ linear code؛ generalized Lee weight | ||
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آمار تعداد مشاهده مقاله: 4 تعداد دریافت فایل اصل مقاله: 2 |
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