Lee Weight and Generalized Lee Weight for Codes Over ‎$‎‎Z_{2^n}$

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Farhang Baftani Farzaneh

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Journal Paper

Paper Information

Journal: Mathematics Interdisciplinary Research Year:2023 | Volume:8 | Issue:1 Page(s): 27-33

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Title

Lee Weight and Generalized Lee Weight for Codes Over ‎$‎‎Z_{2^n}$

Author(s)

Farhang Baftani Farzaneh | Issue Writer Certificate

Keywords

Linear code

Hamming Weight

Generalized Lee Weight

‎$‎(u

u+v)‎$‎- construction of Codes‎

Abstract

‎‎‎‎‎‎Let $‎\mathbb{Z}_m$ be the ring of integers modulo $m$ in which $m=2^n$ for arbitrary $n$‎. ‎In this paper‎, ‎we will obtain a relationship between $wt_L(x)‎, ‎wt_L(y)$ and $wt_L(x+y)$ for any $x‎, ‎y \in ‎\mathbb{Z}_m$‎. ‎‎Let ‎$‎‎d_r^L(C)$‎‎ ‎denote ‎the ‎‎$r‎‎$‎-th Generalized Lee Weight for code $C$ in which ‎$‎‎C$ ‎is ‎a Linear code of length $n$ over $‎\mathbb{Z}_4$‎. Also, ‎suppose that $C_1$ and $ C_2$ are two codes over $‎\mathbb{Z}_4$ and $C$ denotes the $(u‎, ‎u+v)$-construction of them‎. ‎In this paper‎, we will obtain an upper bound for $d_r^L(C)$ for all $r$‎, ‎$1 \leq r \leq rank(C)$‎. In addition, ‎we will obtain $d_1^L(C)$ in terms of $d_1^L(C_1)$ and $d_1^L(C_2)$.

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