8,512 research outputs found
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
Geometrically closed rings
We develop the basic theory of geometrically closed rings as a generalisation
of algebraically closed fields, on the grounds of notions coming from positive
model theory and affine algebraic geometry. For this purpose we consider
several connections between finitely presented rings and ultraproducts, affine
varieties and definable sets, and we introduce the key notion of an arithmetic
theory as a purely algebraic version of coherent logic for rings.Comment: 18 page
On Quadratic Inverses for Quadratic Permutation Polynomials over Integer Rings
An interleaver is a critical component for the channel coding performance of
turbo codes. Algebraic constructions are of particular interest because they
admit analytical designs and simple, practical hardware implementation. Sun and
Takeshita have recently shown that the class of quadratic permutation
polynomials over integer rings provides excellent performance for turbo codes.
In this correspondence, a necessary and sufficient condition is proven for the
existence of a quadratic inverse polynomial for a quadratic permutation
polynomial over an integer ring. Further, a simple construction is given for
the quadratic inverse. All but one of the quadratic interleavers proposed
earlier by Sun and Takeshita are found to admit a quadratic inverse, although
none were explicitly designed to do so. An explanation is argued for the
observation that restriction to a quadratic inverse polynomial does not narrow
the pool of good quadratic interleavers for turbo codes.Comment: Submitted as a Correspondence to the IEEE Transactions on Information
Theory Submitted : April 1, 2005 Revised : Nov. 15, 200
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