tailieunhanh - Báo cáo toán học: "Linear Codes over Finite Chain Rings"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Linear Codes over Finite Chain Rings. | Linear Codes over Finite Chain Rings Thomas Honold Zentrum Mathematik Technische Universitat Miinchen D-80290 Miinchen Germany honold@ Ivan Landjev Institute of Mathematics and Informatics Bulgarian Academy of Sciences 8 Acad. G. Bonchev str. 1113 Sofia Bulgaria ivan@ Submitted December 20 1998 Accepted December 18 1999 AMS Subject Classification Primary 94B27 Secondary 94B05 51E22 20K01. Abstract The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a well-known result for linear codes over fields we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings. 1 Introduction In the past decade a substantial research has been done on linear codes over finite rings. Traditionally authors used to focus their research on codes over integer residue rings especially Z4. Nowadays quite a few papers are concerned with linear codes over other classes of rings cf. . 2 7 11 12 16 17 21 24 42 43 44 50 . THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R11 2 The aim of this paper is to develop the fundamentals of the theory of linear codes over the class of finite chain rings. There are several reasons for choosing this class of rings. First of all it contains rings whose properties lie closest to the properties of finite fields. Hence a theory of linear codes over finite chain rings is expected to resemble the theory of linear codes over finite fields. Secondly the class of finite chain rings contains important representatives like integer residue rings of prime power order and Galois rings. Codes over such rings appeared in various contexts in recent .