Research On The Repeated-root Negacyclic Codes Over The Finite Ring Z_p² (p≠2)

With the development of techniques and the realization of theory of error-correcting codes, error-correcting code technology can be very widely used, such as used in digital communication systems, and computer storage and computing systems, large scale integrated circuit design, even in the neural network, some ideas and methods of decoding of error-correcting codes also be used, which makes more and more scientists pay attention to the error-correcting code theory.However, cyclic codes are considered to be one of the most important subclasses of error-correcting code. There is a long history of the research at cyclic codes over the finite fields. The theoretical achievements and the practical applications are all very rich. Over the last decade, the researchers begin to study the cyclic codes over the finite exchange rings, and a lot of comprehensive researches on the simple root cyclic codes have been got. But it seems to be very difficult to the repeated-root cyclic codes, that's because over a finite exchange ring the polynomial X~n-1 can not be factored uniquely, so the research results are not many enough. And the researches on the negacyclic codes which can be seen as the promotion of the cyclic codes have the same situation.Now, we study the repeated-root negacyclic codes, which are over the finite ring (?)_p~2(p≠2). As follows the details are given:1. We give the generators of negacyclic codes of length p~kn, which are over the finite ring (?)_p~2, and the enumeration formula of the number of such negacyclic codes, by using the discrete Fourier transform.2. Using the Mattson-Solomon polynomial, we derive the generators of the duals of negacyclic codes of length p~kn, which are over the finite ring (?)_p~2. And some properties of self-dual negacyclic codes are also obtained.