The LCD Codes Over F_q+uF_q+u~2F_q

The coding theory was founded by mathematician Shannon,who published a groundbreaking article titled "Mathematical Theory of Communication" in 1948.In the following years,numerous mathematicians created fruitful results in the coding of binary and finite fields,constructing a large number of high-performance classical error correction codes,such as Hamming codes(1950),BCH codes(1 960),LDPC codes(1963),Turbo codes(1990),Polar codes(2008)and so on.These error correcting codes have been widely used in practical production and life such as random-access memory,communication coding technology,and Blu-ray optical discs.Since the 1990s,coding theory has been studied on finite rings,originating from the discovery of some nonlinear binary codes with linear properties,which are the image after undergoing nonlinear mapping(i.e.Gray map)on Z4.At the same time,the Mac Williams theorem is a very important cornerstone of coding theory on finite fields.Wood discovered that the Mac Williams theorem also holds on finite commutative Frobenius rings,thus expanding the research scope of coding from finite fields to finite commutative Frobenius rings.The main focus of this article is the linear complementary dual codes on the ring R=Fq+uFq+u2Fq,u3=u,That is to say,the intersection of linear code and its dual code is{0}.The research method is mainly to convert codes on a ring into codes on a field by defining different forms of Gray maps.For the finite field Fq from ring R,this article mainly studies the differences in the following three situations:(1)we use R1 to represent the situation of q=2,with the help of the code generator matrix,a sufficient condition is given for the codes on the ring to be LCD codes,and an example is provided to illustrate that is only a sufficient condition.By using Gray map,n-length linear codes on R are mapped to n-length linear code on the F2 and the relationship between the linear code on the loop being an LCD code and the linear code by Gray map being an LCD code;(2)we use R2 to represent the situation of q=3,the elements on R are transformed into elements on F3 through primitive idempotents.We discuss and provide the form of the generator matrix.We provide a necessary and sufficient condition for the code on ring R to be an LCD code,and prove that the LCD code on ring R is not equivalent to any non LCD codes.Then,we use Gray map to convert the code on Rn into a quasicyclic code with an index of 3,and construct the constacyclic LCD code on Rn and quasicyclic LCD code on F3;(3)we use R3 to represent the situation of q=pe,p>3,the k-Galois LCD code on ring R is defined,and the necessary and sufficient conditions for the linear code of ring R to be a k-Galois LCD code are given.At the same time,the necessary and sufficient conditions for the linear code of ring R to be a k-Galois LCD code are explored through Gray map.and the necessary and sufficient conditions for the constacyclic k-Galois LCD code on R are discussed,with some examples provided.