The number of fuzzy subgroups and codes with some applications

Any statement in classical logic is true or not, there is no third case. However sometimes even general definite descriptions are not enough to explain things. In order to define these things we need some grades. The new logic with grades of elements is called "Fuzzy Logic". Fuzzy Logic was first born in 500 B.C. with Buddha and also with Aristoteles after 200 years. However the most important scientific study that lightens the studies about fuzzy logic for the last half century is Professor Zadeh's -from the University of California, Berkeley- original paper: "Fuzzy Sets" (Zadeh [19]). This work is the pioneer of the fuzzy studies. Algebraic constructions related with fuzzy is due to A. Rosenfeld (Rosenfeld [1]) and P. S. Das (Das [2]). The word "fuzzy" means blurriness. When talking about the elements of any fuzzy set, we do not use definite expressions such as "an element or not" but "an element with any degree". Beauty, youth, senility, lankiness, diligence, intelligence are some examples of fuzzy expressions since they vary from person to person. On the other hand, Algebraic Coding Theory, one of the most important field of application of algebra, has being studied recently by mathematicians. This theory was first begun with the marvellous paper of Claude Shannon: "A mathematical theory of xvi communication" (Shannon [7]). While in the earlier stages everything was over finite fields, in 1994, by the work (Hammons [8]) of P.V. Kumar and his collaborates, codes over finite rings have been studied. In recent years, codes over some special rings and their properties are popular. Besides, examining linear codes in terms of combinatorial structure, namely finding the number of the subcodes of a linear code, is a really important problem. This problem is completely solved for the codes over finite fields and presented by Gaussian binomial coefficients. On the other hand, a wide range of studies about the number of the codes over rings ([9], [10], [11], [42]) has been done. Hence, in Section 7, linear codes over some finite rings were examined and formulas which makes easier to find their number were obtained. In this work; we give some fundamentals of abstract algebra in Sections 2 and 5, concepts of fuzzy algebra and number of fuzzy subgroups of some Abelian groups in Sections 3 and 4, notions about Algebraic Coding Theory, number of linear codes and some applications in Sections 6, 7 and 8. Key words: Fuzzy subgroups, Equivalence classes, Maximal chains, Algebraic Coding Theory, Gaussian binomial coefficients, Designs, Number sequences