Linear codes over 𝔽4R and their MacWilliams identity

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

DUAL CODE AND MACWILLIAMS IDENTITY ON ADDITIVE CODE

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.

THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES

We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms.

The MacWilliams Identity of Linear Codes over Mn×s(Fp+uFp+vFp+uvFp) with Respect to RT Metric

The definition of the exact complete ρ weight enumerator over Mn×s(Fp+uFp+vFp+uvFp) is given, and the MacWilliams identity with respect to RT metric for the exact complete ρ weight enumerator of linear codes over Mn×s(Fp+uFp+vFp+uvFp) is obtained. Finally, a example is presented to illustrate the obtained results.