A [7 4 2] Linear Code Due to the Cayley Table for for the Generated Points of the (123)/(132) - Avoiding Patterns of the Non – Associative AUNU Schemes
DOI:
https://doi.org/10.62054/ijdm/0204.08Keywords:
Cayley tables, AUNU Scheme, Hamming codes, Standard Generator matrix, Reduced row Echelon formAbstract
In this communication, we present the construction of a [7 4 2] - linear code which is an extended code of the [ 6 4 1 ] code and is in one-to-one correspondence with the known [ 7 4 3 ] - Hamming code. Our construction is due to the Cayley table for of the generated points of as permutations of the -avoiding patterns of the non-associative AUNU schemes as reported by the authors earlier. First, the Cayley table for is converted to a matrix say in the binary system using arithmetic. The matrix which is of order is then shown to generate a linear code of size. Next, echelon row operations are used to transform into a standard generator matrix say , for a [ 6 4 1 ] code. Finally, the [6 4 1] code is extended to obtain the desired [7 4 2]-linear Code which is in one-to-one correspondence with the known [ 7 4 3 ]- Hamming code.
References
REFERENCES:
Chun P.B, Ibrahim A.A, and Garba A.I, (2016a) Algebraic theoretic properties of the avoiding class of AUNU permutation patterns: Application in the generation and analysis of linear codes. International Organization for Scientific Research(IOSR), Journal of Mathematics12(1) pp 1-3.
Chun P. B , Ibrahim A.A , Garba A.I (2016b). Algebraic Theoretic Properties of the Non-associative Class of (132)-Avoiding Patterns of AUNU Permutations: Applications in the Generation and Analysis of a General Cyclic Code. Computer Science and Information Technology, 4 , 45 - 47. doi: 10.13189/csit.2016.040201.
David, .J & John-Lark, .K (2011). Selected unsolved problems in Coding theory . Springer Newyork Dordrecht Heidelberg London.
Hoffman, D.G., Leonard, D.A., Lindner, C.C., Phelps, K.T., Rodger, C.A., & Wall, J.R (1992), Coding Theory: The Essentials. Mercel Dekker, Inc, 1992.
Ibrahim M, Ibrahim A.A, Yakubu, M.A.(2012) Algebraic theoretic properties of the (132)-Avoiding class of Aunu patterns application in Eulerian graphs, Journal of Science and Technology Research, 2012.
Ibrahim A.A., Chun P.B., Garba A.I. and Abubakar S.I. (2016). A standard Generator/Parity Check Matrix for Codes from the Cayley tables due to the Non-associative (123)-Avoiding Patterns of AUNU numbers. (2016) Universal Journal of Applied Mathematics VOL. 4(2) pp 39-41 Doi:10.13189/ujam.2016.040202
Ibrahim A.A., and Abubakar S.I.(2016). Non-Associative Property of 123-Avoiding Class of Aunu Permutation Patterns, Advances in Pure Mathematics, 6(2)(2016), 51-57, http://dx.doi.org/10.4236/apm.2016.62006
Usman A. & Ibrahim A.A (2011) A new generating function for AUNU patterns: Application in integer Group Modulo n . Nigerian Journal of Basic and Applied Sciences 19 (1).
Vasantha W.b, Florentin .S., & Ilanthenral .K. (2010) Supper special Codes Using Supper Matrices. Infolearnquest, Ann Arbor, 2010.
Willem, H. H (2011) Matrices for graphs, Designs and Codes, (NATO Science for Peace and security, series 29 Information Security), Coding theory and Related Combinatorics, IOS press, 2011.
Downloads
Published
Data Availability Statement
See references for research data
Issue
Section
License
Copyright (c) 2025 Chun B. Pamson (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors are solely responsible for obtaining permission to reproduce any copyrighted material contained in the manuscript as submitted. Any instance of possible prior publication in any form must be disclosed at the time the manuscript is submitted and a
copy or link to the publication must be provided.
The Journal articles are open access and are distributed under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 IGO License, which permits use,
distribution, and reproduction in any medium, provided the original work is properly cited.
No modifications or commercial use of the articles are permitted.
