# The ideal entropy of BCI-algebras and its application in the binary linear codes

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## 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 *C*^{c}. 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.

## Keywords

Ideal entropy BCI-algebra Linear code Quotient BCI-algebra Decoding Equivalent linear codes## Notes

### Acknowledgements

The authors are very indebted to the referees for valuable suggestions that improved the readability of the paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Bennett CH, Li M, Ma B (2003) Chain letters and evolutionary histories. Sci Am Am Ed 288(6):76–81CrossRefGoogle Scholar
- Berlekamp ER (1968) Algebraic coding theory, vol 129. McGraw-Hill, New YorkzbMATHGoogle Scholar
- 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). IEEEGoogle Scholar
- Dikranjan D et al (2009) Algebraic entropy for abelian groups. Trans Am Math Soc 361(7):3401–3434MathSciNetCrossRefzbMATHGoogle Scholar
- Hao J, Li CX (2004) On ideals of an ideal in a BCI-algebra. Sci Math Japonicae 60(3):627MathSciNetzbMATHGoogle Scholar
- Huang Y (2006) BCI-algebra. Elsevier, AmsterdamGoogle Scholar
- Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42(1):26–29MathSciNetCrossRefzbMATHGoogle Scholar
- Ling S, Solé P (2001) On the algebraic structure of quasi-cyclic codes. I. Finite fields. IEEE Trans Inf Theory 47(7):2751–2760MathSciNetCrossRefzbMATHGoogle Scholar
- Ling S, Xing C (2004) Coding theory: a first course. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Marcolli M, Tedeschi N (2015) Entropy algebras and Birkhoff factorization. J Geom Phys 97:243–265MathSciNetCrossRefzbMATHGoogle Scholar
- Mehrpooya A, Ebrahimi M, Davvaz B (2016) The entropy of semi-independent hyper MV-algebra dynamical systems. Soft Comput 20(4):1263–1276CrossRefzbMATHGoogle Scholar
- Nemzer LR (2017) Shannon information entropy in the canonical genetic code. J Theor Biol 415:158–170CrossRefzbMATHGoogle Scholar
- Parkash O, Kakkar P (2014) New measures of information and their applications in coding theory. Can J Pure Appl Sci 8(2):2905–2912Google Scholar
- Van Lint JH (2012) Introduction to coding theory, vol 86. Springer, BerlinzbMATHGoogle Scholar
- Walters P (2000) An introduction to ergodic theory, vol 79. Springer, BerlinzbMATHGoogle Scholar