Quasi-cyclic Codes Over Finite Chain Rings

Quasi-cyclic codes form a remarkable generalization of cyclic codes. They are asymptotically good as they meet a modified version of Gibert-Varshamov bound . They are closely linked to convolutional codes. Recently there has been a great interest for their applications in studying Turbo codes and many Low-Density Parity Check (LDPC) codes. It is well known that Turbo codes and LDPC codes have been proved to be capacity approaching codes.In the 1980s, the theory of error-correcting codes over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes over Z 4under the Gray map. Since then, codes over finite rings have been given more attention, however, there has been a limited study on quasi-cyclic codes over rings. In this paper, we study structural properties and enumeration of quasi-cyclic Codes over finite chain rings. We also study that a quasi-cyclic code of over Fp + uFp + + u kFp is uniquely equivalent to a quasi-cyclic code of over Fp . The details are given as follows:1. We give a decomposition of quasi-cyclic codes over GR ( p s,n ) into quasi-cyclic codes over some Galois extention rings of GR ( p s,n ), and give a general form of generator of a 1-generator quasi-cyclic codes over GR ( p s,n ),define a conception of a free quasi-cyclic codes , give a general form of generator and rank of a free 1-generator quasi-cyclic codes over GR ( p s,n ),and a condition of a free quasi-cyclic codes to be equivalent to its duality is given,and show some properties of 1-generator quasi-negacyclic codes over GR ( 2 s,n ).2. We show quasicyclic codes of length n = ml and index l over the Z ps to be isomorphic to the [ ] 1Z px x ? -submodules of GR ( p s , l )[ x ]x m? 1, define the concept of free quasicyclic codes, show the relation between 1-generator free quasicyclic codes and multi-generator free quasicyclic codes. Using the module representation of quasicyclic codes, we determine the dimension of 1-generator free quasicyclic codes and multi-generator free quasicyclic codes and give their algebraic lower bound on minimum distance.3. We show that there exists a unique dual basis of a free basis of GR ( p s,n ) over Z ps is shown.,there exists a one-to-one correspondence between the 1-generator quasicyclic codes over Z ps with p.c.p h(x) and a class of special elements of the ideal in GR ( p s , n )[ x ]x m? 1, and give the enumeration formula of 1-generator quasicyclic codes over Z ps with p.c.p h(x) .4. We define ,let R = Fp + uFp + + u kFp,the Gray map from R n1to p kn1Fp ,and give the proposition of this map, and show that a quasi-cyclic code of length 1n = n pswith index p st over is uniquely equivalent to a quasi-cyclic code of length n = n pswith index p st over Fp ,where ( )t | n1 , n1 , p = 1.Then the quasi-cyclic code over the ring R can be taken as the quasi-cyclic code over the ring Fp .