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

Document Type : Original Scientific Paper

Author

Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

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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References

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Mathematics Interdisciplinary Research
Volume 8, Issue 1
Special Issue: Proceedings of the 27th Iranian Algebra Seminar (IAS27) --- Editors: Reza Sharafdini and Mojtaba Sedaghatjoo
March 2023
Pages 27-33

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