AN ALTERED GROUP RING CONSTRUCTION OF THE [24, 12, 8] AND [48, 24, 12] TYPE II LINEAR BLOCK CODE
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders $2,3,4,$ and $5$, and by applying the construction over the binary field and the ring $F_{2}+uF_{2}$, we obtain extremal binary self-dual codes of various lengths: $12, 16, 20, 24, 32, 40,$ and $48$. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code $[24,12,8]$ and the unique Extended Quadratic Residue $[48,24,12]$ Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in \cite{Kaya2020} by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.
Abstract
In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders $2,3,4,$ and $5$, and by applying the construction over the binary field and the ring $F_{2}+uF_{2}$, we obtain extremal binary self-dual codes of various lengths: $12, 16, 20, 24, 32, 40,$ and $48$. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code $[24,12,8]$ and the unique Extended Quadratic Residue $[48,24,12]$ Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in \cite{Kaya2020} by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.
참고문헌 (16)
- E.F. Assmus, "New 5-designs", Journal of Combinatorial Theory, 6(2), pp.122~151 (1969), Elsevier BV[DOI]
- Frank Bernhardt, "The extended golay codes considered as ideals", Journal of Combinatorial Theory, Series A, 55(2), pp.235~246 (1990), Elsevier BV[DOI]
- Steven T. Dougherty, "Springer-Briefs in Mathematics" (2017), Springer
- Steven T. Dougherty, "Self-dual codes over commutative Frobenius rings", Finite Fields and Their Applications, 16(1), pp.14~26 (2010), Elsevier BV[DOI]
- Steven T. Dougherty, "Codes over Rk, Gray maps and their binary images", Finite Fields and Their Applications, 17(3), pp.205~219 (2011), Elsevier BV[DOI]
- S. Houghten, "Construction of (48, 24, 12) doubly-even self-dual codes", Congr. Numer, 103, pp.41~53 (1994)
- S.K. Houghten, "The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code", IEEE Transactions on Information Theory, 49(1), pp.53~59 (2003), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- W. C. Huffman, "Fundamentals of Error-Correcting Codes" (2003), Cambridge University Press
- T. Hurley, "Group rings and rings of matrices", Int. J. Pure Appl. Math, 31(3), pp.319~335 (2006)
- Joe Gildea, "Constructing self-dual codes from group rings and reverse circulant matrices", Advances in Mathematics of Communications, 15(3), pp.471~ (2021), American Institute of Mathematical Sciences (AIMS)[DOI]
- Joe Gildea, "Double bordered constructions of self-dual codes from group rings over Frobenius rings", Cryptography and Communications, 12(4), pp.769~784 (2020), Springer Science and Business Media LLC[DOI]
- type II linear block code, "Designs, Codes and Cryptography", 2011, 29~41, pp.Springer Science and Business Media LLC, https:
- Ian McLoughlin, "A Group Ring Construction of the Extended Binary Golay Code", IEEE Transactions on Information Theory, 54(9), pp.4381~4383 (2008), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- C. Polcino Milies, "Algebra and Applications", pp.1~ (2002), Kluwer Academic Publishers
- E.M. Rains, "Shadow bounds for self-dual codes", IEEE Transactions on Information Theory, 44(1), pp.134~139 (1998), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- "SAGE: Mathematical software, version 2.10"[URL]