Z_pZ_p² Additive Codes And Their Applications

This article aims to solve three problems. One is the enumeration of additive codes, the second is the theory of p2 pZ Z additive codes and the last is some considerations based on the error-correcting capabilities of codewords.Firstly, we solve the enumeration of additive codes, including four aspects of problems, that is, the enumeration formulae of any type of additive codes in any finitely abelian p-groups, the numerical and structural relations of the same type of additive codes in the different finitely abelian p-groups, the enumeration formulae of any order of additive codes in any finitely abelian p-groups, the numerical and structural relations of the same order of additive codes in the different finitely abelian p-groups.Secondly, we study the generator matrix, parity-check matrix, systematic encoding and permutation decoding of p2 pZ Z additive codes, respectively, mainly generalizing the relevant theory of 2 4Z Z additive codes. We will not only generalize the conclusions, but also the methods of 2 4Z Z additive codes.The last part of our paper considers a practical problem, that is, to what extent could the error-correcting capabilities of a code and its codewords represent the error-correcting capabilities of the whole code, respectively. We first distinguish the error-correcting capabilities of codewords fron the error-correcting capabilities of the code they belong to, and then give the concepts of bound and the perfect code. The main results of this part are two kinds of optimizing algorithms for codes in order to increase the minimum distance and the number of codewords of a code respectively on the basis of the minimum distances of codewords. As an application of the algorithm for increasing the minimum distance of a code, we obtain another construction of binary Hamming code. Lastly we briefly describe the encoding and decoding on the basis of the error-correcting capabilities of codewords.