General theory of doubly periodic arrays over an arbitrary finite field and its applications

Abstract
A general theory of doubly periodic (DP) arrays over an arbitrary finite field GF(q)is presented. First the basic properties of DP arrays are examined. Next modules of linear recurring (LR) arrays are defined and their algebraic properties discussed in connection with ideals in an extension ring\tilde{R}of the ringRof bivariate polynomials with coefficients in GF(q). A finite\tilde{R}-module of DP arrays is shown to coincide with the\tilde{R}-module of LR arrays dermed by a zero-dimensional ideal in\tilde{R}. Equivalence relations between DP arrays are explored, i.e., rearrangements of arrays by means of unimodular transformations. Decimation and interleaving of arrays are defined in a two-dimensional sense. The general theory is followed by application to irreducible LR arrays. Among irreducible arrays,M-arrays are a two-dimensional analog ofM-sequences and may be constructed fromM-sequences by means of unimodular transformations. The results of this paper are also important in studying properties of Abelian codes.

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