To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras asssociated to layered graphs.
In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs.
The linear complexity of an /V-periodic sequence with components in a field of characteristic p, where A' = np" and gcd(ra,p) = 1, is characterized in terms of the n l ' roots of unity and their multiplicities as zeroes of the polynomial whose cofficients are the first N digits of the sequence. Hasse derivatives are then introduced to quantify these multiplicities and to define a new generalized discrete Fourier transform that can be applied to sequences of arbitrary length A r with components in a field of characteristic p, regardless of whether or not gcd(A r , p) = 1. This generalized discrete Fourier transform is used to give a simple proof of the validity of the well-known Games-Chan algorithm for finding the linear complexity of an /V-periodic binary sequence with N = f l v and to generalize this algorithm to apply to A r-periodic sequences with components in a finite field of characteristic p when N = p v. It is also shown how to use this new transform to study the linear complexity of Hadamard (i.e., component-wise) products of sequences.
A method for analyzing the linear complexity of nonlinear filterings of PN-sequences that is based on the Discrete Fourier Transform is presented.The method makes use of "Blahut's theorem", which relates the linear complexity of an N-periodic sequence in G P ( P )~ and the Hamming weight of its frequency-domain associate. To illustrate the power of this approach, simple proofs are given of Key's bound on linear complexity and of a generalization of a condition of Groth and Key for which equality holds in this bound.
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