Abstract
This paper defines the concept of ideal entropy for BCI-algebras in general, and it tries to describe some of its properties. Moreover, the present study will show that \( F_{2}^{n} \) (i.e., sets of every binary code word of length n) is a BCI-algebra, and that each ideal of \( F_{2}^{n} \) is a linear code. The present study defines the concept of cosets by using the quotient BCI-algebra \( \frac{{F_{2}^{n} }}{I} \) and obtains their properties. This study defines the complement of a linear code, which is itself a linear code, which is denoted by the symbol Cc. Further, the present study defines the standard complement of a linear code, which is unique. This study proves that each equivalence class in \( F_{2}^{n} /C^{c} \) contains one and only one code word of the linear code C. This property can be used for decoding. Finally, the present study shows that two linear codes are equivalent if and only if they have the same ideal entropy.
Similar content being viewed by others
References
Bennett CH, Li M, Ma B (2003) Chain letters and evolutionary histories. Sci Am Am Ed 288(6):76–81
Berlekamp ER (1968) Algebraic coding theory, vol 129. McGraw-Hill, New York
Castellano I, Bruno AG (2017) Topological entropy for locally linearly compact vector spaces. arXiv preprint, arXiv:1701.00676
Da Silva A (2016) Affine systems on Lie groups and invariance entropy. In: 2016 IEEE 55th conference on decision and control (CDC). IEEE
Dikranjan D et al (2009) Algebraic entropy for abelian groups. Trans Am Math Soc 361(7):3401–3434
Hao J, Li CX (2004) On ideals of an ideal in a BCI-algebra. Sci Math Japonicae 60(3):627
Huang Y (2006) BCI-algebra. Elsevier, Amsterdam
Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42(1):26–29
Ling S, Solé P (2001) On the algebraic structure of quasi-cyclic codes. I. Finite fields. IEEE Trans Inf Theory 47(7):2751–2760
Ling S, Xing C (2004) Coding theory: a first course. Cambridge University Press, Cambridge
Marcolli M, Tedeschi N (2015) Entropy algebras and Birkhoff factorization. J Geom Phys 97:243–265
Mehrpooya A, Ebrahimi M, Davvaz B (2016) The entropy of semi-independent hyper MV-algebra dynamical systems. Soft Comput 20(4):1263–1276
Nemzer LR (2017) Shannon information entropy in the canonical genetic code. J Theor Biol 415:158–170
Parkash O, Kakkar P (2014) New measures of information and their applications in coding theory. Can J Pure Appl Sci 8(2):2905–2912
Van Lint JH (2012) Introduction to coding theory, vol 86. Springer, Berlin
Walters P (2000) An introduction to ergodic theory, vol 79. Springer, Berlin
Acknowledgements
The authors are very indebted to the referees for valuable suggestions that improved the readability of the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by A. Di Nola.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ebrahimi, M., Izadara, A. The ideal entropy of BCI-algebras and its application in the binary linear codes. Soft Comput 23, 39–57 (2019). https://doi.org/10.1007/s00500-018-3620-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-018-3620-0