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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 ℤ4, 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 ℤ4 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 wtL(D) and called the Lee Support Weight of D. In addition, the m-th Generalized Lee Weight (GLW) for D, denoted by dmL(D), is defined as the minimum of the Lee Support Weights of all submodules of D of rank m. In other words, dmL(D) = min{wtL(E) ; E is a ℤ4-submodule of D, rank(E) = m}. It is obtained that for m, 1 ≤ m ≤ rank(D), we have ⌊(dmL(D) - 2m + 1) / 2⌋ ≤ l - rank(D). The code which meets the recent upper bound is called Maximum Lee Distance separable with respect to Rank (m-th MLDR) code. Also, if dmH(D) denotes the m-th GHW for code D, it is defined as dmH(D) = min{|supp(E)| ; E is a ℤ4-submodule of D and rank(E) = m}. The upper bound for dmH(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 detail. | ||
| کلیدواژه‌ها | ||
| Lee weight؛ Hamming weight؛ MHDR code؛ linear code؛ generalized Lee weight | ||
| مراجع | ||
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[1] S. T. Dougherty, K. Shiromoto, MDR codes over Zk , IEEE Trans. Inform. Theory 46(1) (2000) 265–269. https://doi.org/10.1109/18.817524 [2] S. T. Dougherty, M. K. Gupta, K. Shiromoto, On generalized weights for codes over Zk , Australian Journal of Combinatorics 31 (2005) 231–248. https://ajc.maths.uq.edu.au/pdf/31/ajc_v31_p231.pdf [3] F. Farhang Baftani, GHW for (x, x + y)-construction of codes, Advanced Studies: Euro-Tbilisi Mathematical Journal 16(1) (2023) 1–6. https://doi.org/10.32513/asetmj/1932200823101 [4] F. Farhang Baftani, Lee weight and generalized Lee weight for codes over Z2 n , Mathematics Interdisciplinary Research 8(1) (2023) 27–33. https://doi.org/10.22052/mir.2023.246385.1348 [5] H. Horimoto, K. Shiromoto, On generalized Hamming weights for codes over finite chain rings, Lecture Notes in Computer Science 2227 (2001) 141–150. https://doi.org/10.1007/3-540-45624-4_15 [6] S. Ling, C. Xing, Coding Theory: A First Course, Cambridge Univ. Press, Cambridge, 2004. [7] J. H. Van Lint, Introduction to Coding Theory, Springer-Verlag, Berlin, 1999. https:// doi.org/ 10.1007/ 978-3-642-58575-3 [8] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37(5) (1991) 1412–1418. https://doi.org/10.1109/18.133259 [9] B. Yildiz, Z. Odemis Ozger, A generalization of the Lee weight to Zp k , TWMS J. App. Eng. Math 2(2) (2012) 145–153. https://dergipark.org.tr/tr/download/article-file/1181205 | ||
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