Original Research Article

Article volume = 2021 and issue = 1

Pages: 44–49

Article publication Date: November, 1, 2021

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# Lee Weight for Direct Sum of Codes

#### Farzaneh Farhang Baftani

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

##### Abstract:

Let C be a linear code of length n over Z4. The Lee support weight of C, denoted by wtL(C), is the sum of Lee weights of all columns of A(C) that A(C) is the jCj n array of all code words in C. For 1 6 r 6 rank(C), the r-th generalized Lee weight with respect to rank (GLWR) for C, denoted by dLr (C), is defined the minimum of all Lee weights of Z4-submodules of C with rank = r. In other words dLr (C) = minfwtL(D);D is a Z4 - submodule of C with rank(D) = rg For linear codes C1 and C2 over Z4 of length n1 and n2, respectively, the Direct Sum of them ,denoted by C1 C2 , is defined as follows: C1 C2 = f(c1, c2) : c1 2 C1, c2 2 C2g. Motivated by finding dLr (C1 C2) in terms of dLr (C1) and dLr (C2), we investigated dLr (C1 C2) and we obtained dLr (C1 C2) for r = 1, 2. Moreover, we generally obtained an upper bound for dLr (C1 C2) for all r, 1 6 r 6 rank(C1 C2).

##### Keywords:

Linear code, Hamming Weight, Lee Weight, Generalized Lee Weight, Direct Sum of Codes.

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##### Cite this article as:

- Farzaneh Farhang Baftani, Lee Weight for Direct Sum of Codes, Communications in Combinatorics, Cryptography & Computer Science, 2021(1), PP.44–49, 2021
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