Soft Computing

, Volume 23, Issue 1, pp 39–57 | Cite as

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

  • Mohammad Ebrahimi
  • Alireza IzadaraEmail author


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.


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



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.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics and ComputerShahid Bahonar University of KermanKermanIran

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