Cyclic Codes Over Ring F₂+uF₂

Cyclic codes are one of the most important class of linear codes. They have strict algebra structure, whose character is easy to analyze; They also have cyclic character, such that encoding and decoding are easy to realize, and have attracted many scholars' attention. In 1957, Prange first began to study cyclic codes over field GF(q), and then the study of cyclic codes over GF (q) was made great progress both on theory and on practice. Recently, the study of cyclic codes was extended from over a field to over a ring. So far, the known results of cyclic codes over rings are limited.Consider cyclic codes over a quarternary ring. There are four different commutative quarternary rings with identity: Galois field F₄, Z₄, F₂ + uF₂ and F₂ + vF₂. In 1996, V. Pless and Z. Qian discussed the cyclic codes of odd length over Z₄; In 1998, A. Bonnecase and P. Udaya discussed the cyclic codes of odd length over F₂ + uF₂; In 2003, T. Blackford discussed the cyclic codes of oddly even length over Z₄; In 2003, T. Abualrub and R. Oehmke discussed the cyclic codes of length 2^e over Z₄. In this thesis, cyclic codes of length 2^e over ring F₂ + uF₂ are studied.This thesis is composed of following sections: In the first section, we present the needed terms and a survey of cyclic codes over rings; In the second section, we discuss the cyclic codes of length 2^e over ring F₂ + uF₂; In the third section, we discuss the dual codes of cyclic codes of length 2^e over ring F₂ + uF₂; In the last section, we obtain two binary codes by Gray map: residue code and torsion code.