10,647 research outputs found
Continuants and some decompositions into squares
In 1855 H. J. S. Smith proved Fermat's two-square using the notion of
palindromic continuants. In his paper, Smith constructed a proper
representation of a prime number as a sum of two squares, given a solution
of , and vice versa. In this paper, we extend the use of
continuants to proper representations by sums of two squares in rings of
polynomials on fields of characteristic different from 2. New deterministic
algorithms for finding the corresponding proper representations are presented.
Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic
forms.Comment: 21 page
Efficient Integer Coefficient Search for Compute-and-Forward
Integer coefficient selection is an important decoding step in the
implementation of compute-and-forward (C-F) relaying scheme. Choosing the
optimal integer coefficients in C-F has been shown to be a shortest vector
problem (SVP) which is known to be NP hard in its general form. Exhaustive
search of the integer coefficients is only feasible in complexity for small
number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz
(LLL) lattice reduction algorithm only find a vector within an exponential
factor of the shortest vector. An optimal deterministic algorithm was proposed
for C-F by Sahraei and Gastpar specifically for the real valued channel case.
In this paper, we adapt their idea to the complex valued channel and propose an
efficient search algorithm to find the optimal integer coefficient vectors over
the ring of Gaussian integers and the ring of Eisenstein integers. A second
algorithm is then proposed that generalises our search algorithm to the
Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed
algorithms are evaluated through simulations and theoretical analysis.Comment: IEEE Transactions on Wireless Communications, to appear.12 pages, 8
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Incremental and Transitive Discrete Rotations
A discrete rotation algorithm can be apprehended as a parametric application
from \ZZ[i] to \ZZ[i], whose resulting permutation ``looks
like'' the map induced by an Euclidean rotation. For this kind of algorithm, to
be incremental means to compute successively all the intermediate rotate d
copies of an image for angles in-between 0 and a destination angle. The di
scretized rotation consists in the composition of an Euclidean rotation with a
discretization; the aim of this article is to describe an algorithm whic h
computes incrementally a discretized rotation. The suggested method uses o nly
integer arithmetic and does not compute any sine nor any cosine. More pr
ecisely, its design relies on the analysis of the discretized rotation as a
step function: the precise description of the discontinuities turns to be th e
key ingredient that will make the resulting procedure optimally fast and e
xact. A complete description of the incremental rotation process is provided,
also this result may be useful in the specification of a consistent set of
defin itions for discrete geometry
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