Structures of commutative linear representation systems and an efficient coding for two‐dimensional arrays

Abstract

Previously, we posted the following question concerning the array realization problem: Assume that one or more (mathematical) model realizes (faithfully describes) an arbitrary (two dimensional) array; if two such models exist that realize the array, then are these models isomorphic? The solution to this question is the array realization theorem: Canonical commutative linear representative systems necessarily exist that realize an arbitrary array, and these systems are identical excluding isomorphism. An investigation of finite dimensional canonical commutative linear representation systems that can be dealt with on a computer is performed. An algorithm is provided that realizes any given finite dimensional array. In this paper, an efficient coding (communications channel coding) is described for arrays that result from the structure problem of commutative linear representative systems based on these results and its solution. This structure problem is an extension of the structure problem occurring in linear system theory to a structure theorem for commutative linear representative systems. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(3): 58–68, 1999