Groebner basis and linear recursive arrays over a ring

Author: Lu Peizhong
Publisher:
Publish Date: 2002-10-01
Features: This book establishes the theory of linear recursive arrays over commutative rings (especially QP rings) using exchange algebra, homological algebra, and Gr?bner bases, and applies this theory to related information technology fields such as error-correcting codes, signal analysis, and cryptanalysis. The book presents a fundamental duality theorem between the array zeroizing module ZerM(I) of a polynomial ideal I and HomR(R[X]/I, R), thereby constructing a generating set for ZerM(I). From this, it further determines the necessary and sufficient condition for the functors ZerM and AnnR[X] to form an inverse Galois correspondence, thus obtaining the zero-point theorem for any ideal in the polynomial ring R[X] over the QF ring R. The form and effectiveness of this theorem are similar to those of the Hilbert Nullstellensatz theorem, making it fundamental and crucial in the study of LRA theory. The book provides a concise discriminant formula for I to be a characteristic ideal of an LRA over a field F, and gradually extends this formula to QF rings. This resolves the open problem proposed by Nechaev and reveals the structure of high-dimensional cyclic codes over QF rings. The book also discusses the important applications of Gr?bner bases in algebraic coding, particularly in decoding areas such as cyclic codes and algebraic geometry. It clearly reveals the precise connection between each element in the minimal Gr?bner basis of a homogeneous characteristic ideal of a finite LRS and each step in the Berlekamp-Massey sequence synthesis algorithm, as well as the cyclic module structure of high-dimensional cyclic codes over the ring.