### DUAL CODE AND MACWILLIAMS IDENTITY ON ADDITIVE CODE

https://doi.org/10.22146/jmt.34471

Miftah Yuliati(1*), Sri Wahyuni(2), Indah Emilia Wijayanti(3)

(2) Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada
(3) Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada
(*) Corresponding Author

#### Abstract

Additive code is a generalization of linear code. It is defined as subgroup of a finite Abelian group. The definitions of Hamming distance, Hamming weight, weight distribution, and homogeneous weight distribution in additive code are similar with the definitions in linear code. Different with linear code where the dual code is defined using inner product, additive code using theories in group to define its dual code because in group theory we do not have term of inner product. So, by this thesis, the definitions of dual code in additive code will be discussed. Then, this thesis discuss about a familiar theorem in dual code theory, that is MacWilliams Identity. Next, this thesis discuss about how to proof of MacWilliams Identity on adiitive code using dual codes which are defined.

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#### References

Delsarte, P., 1972, Bounds for Unrestricted Codes, by Linear Programming, Philips Res. Rep. 27 (1972), 272-289. MR MR0314545.

Rio, A. D., 2017, Codes Over Rings and Modules, http://www.di.univr.it/dovumenti/Occorrenzalns/mat did/matdid553467.pdf, diakses 25 Januari 2017.

Szabo, S. dan Wood, J. A., 2016 ,Properties of Dual Codes Defined by Nondegenerate Forms, Jacodes Math.

Wood, J. A., 2009, Lecture Notes on Dual Codes and The MacWilliams Identities, Amer. J Math.

DOI: https://doi.org/10.22146/jmt.34471

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