1,206 research outputs found
On bi-unitary perfect polynomials over
We give all bi-unitary non splitting even perfect polynomials over the prime
field of two elements, which are divisible by Mersenne irreducible polynomials
raised to special exponents. We also identify all bi-unitary perfect
polynomials over the same field, with at most four irreducible factors. We then
complete, in this manner, a list given by J.T. B. Beard Jr
Fixed points of the sum of divisors function on ({{mathbb{F}}}_2[x])
We work on an analogue of a classical arithmetic problem over polynomials. More precisely,
we study the fixed points (F) of the sum of divisors function (sigma : {mathbb{F}}_2[x] mapsto {mathbb{F}}_2[x])
(defined mutatis mutandi like the usual sum of divisors over the integers)
of the form (F := A^2 cdot S), (S) square-free, with (omega(S) leq 3), coprime with (A), for (A) even, of whatever degree, under some conditions. This gives a characterization of (5) of the (11) known fixed points of (sigma) in ({mathbb{F}}_2[x])
Algebraic techniques in designing quantum synchronizable codes
Quantum synchronizable codes are quantum error-correcting codes that can
correct the effects of quantum noise as well as block synchronization errors.
We improve the previously known general framework for designing quantum
synchronizable codes through more extensive use of the theory of finite fields.
This makes it possible to widen the range of tolerable magnitude of block
synchronization errors while giving mathematical insight into the algebraic
mechanism of synchronization recovery. Also given are families of quantum
synchronizable codes based on punctured Reed-Muller codes and their ambient
spaces.Comment: 9 pages, no figures. The framework presented in this article
supersedes the one given in arXiv:1206.0260 by the first autho
The effect of convolving families of L-functions on the underlying group symmetries
L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M -->
oo, the statistical behavior (1-level density) of the low-lying zeros of
L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of
matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity,
where each G_i is one of the classical compact groups (unitary, symplectic or
orthogonal). Assuming that the convolved families of L-functions F_N x G_M are
automorphic, we study their 1-level density. (We also study convolved families
of the form f x G_M for a fixed f.) Under natural assumptions on the families
(which hold in many cases) we can associate to each family L of L-functions a
symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding
low-lying zero statistics agree with those of the unitary (resp., symplectic or
orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry
type of the convolved family is the product of the symmetry types of the two
families. A similar statement holds for the convolved families f x G_M. We
provide examples built from Dirichlet L-functions and holomorphic modular forms
and their symmetric powers. An interesting special case is to convolve two
families of elliptic curves with rank. In this case the symmetry group of the
convolution is independent of the ranks, in accordance with the general
principle of multiplicativity of the symmetry constants (but the ranks persist,
before taking the limit N,M --> oo, as lower-order terms).Comment: 41 pages, version 2.1, cleaned up some of the text and weakened
slightly some of the conditions in the main theorem, fixed a typ
Cyclone Codes
We introduce Cyclone codes which are rateless erasure resilient codes. They
combine Pair codes with Luby Transform (LT) codes by computing a code symbol
from a random set of data symbols using bitwise XOR and cyclic shift
operations. The number of data symbols is chosen according to the Robust
Soliton distribution. XOR and cyclic shift operations establish a unitary
commutative ring if data symbols have a length of bits, for some prime
number . We consider the graph given by code symbols combining two data
symbols. If such random pairs are given for data symbols, then a
giant component appears, which can be resolved in linear time. We can extend
Cyclone codes to data symbols of arbitrary even length, provided the Goldbach
conjecture holds.
Applying results for this giant component, it follows that Cyclone codes have
the same encoding and decoding time complexity as LT codes, while the overhead
is upper-bounded by those of LT codes. Simulations indicate that Cyclone codes
significantly decreases the overhead of extra coding symbols
Quantum Hypothesis Testing with Group Structure
The problem of discriminating between many quantum channels with certainty is
analyzed under the assumption of prior knowledge of algebraic relations among
possible channels. It is shown, by explicit construction of a novel family of
quantum algorithms, that when the set of possible channels faithfully
represents a finite subgroup of SU(2) (e.g., ) the
recently-developed techniques of quantum signal processing can be modified to
constitute subroutines for quantum hypothesis testing. These algorithms, for
group quantum hypothesis testing (G-QHT), intuitively encode discrete
properties of the channel set in SU(2) and improve query complexity at least
quadratically in , the size of the channel set and group, compared to
na\"ive repetition of binary hypothesis testing. Intriguingly, performance is
completely defined by explicit group homomorphisms; these in turn inform simple
constraints on polynomials embedded in unitary matrices. These constructions
demonstrate a flexible technique for mapping questions in quantum inference to
the well-understood subfields of functional approximation and discrete algebra.
Extensions to larger groups and noisy settings are discussed, as well as paths
by which improved protocols for quantum hypothesis testing against structured
channel sets have application in the transmission of reference frames, proofs
of security in quantum cryptography, and algorithms for property testing.Comment: 22 pages + 9 figures + 3 table
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