Permutation polynomials on matrices

AbstractFamilies of examples are presented of polynomials over a finite field or a residue class ring of the integers, which, on substitution, permute the n×n matrices over that field or residue class ring

Spectra of phase point operators in odd prime dimensions and the extended Clifford group

We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.Comment: Latex, 19page

A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Pl\"ucker embedding of these codes and show how the orbit structure is preserved in the embedding.Comment: submitted to Designs, Codes and Cryptograph

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

We present two methods for the construction of quantum circuits for quantum error-correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n-k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC

Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

We relate the construction of a complete set of cyclic mutually unbiased bases, i. e., mutually unbiased bases generated by a single unitary operator, in power-of-two dimensions to the problem of finding a symmetric matrix over F_2 with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form 2^(2^k) we present a solution that shows an analogy to an open conjecture of Wiedemann in finite field theory. Finally, we discuss the equivalence of mutually unbiased bases.Comment: 11 pages, added chapter on equivalenc

Pure Asymmetric Quantum MDS Codes from CSS Construction: A Complete Characterization

Using the Calderbank-Shor-Steane (CSS) construction, pure $q$-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.Comment: Change in authors' list. Accepted for publication in Int. Journal of Quantum Informatio

Detection of Salmonella in Poultry Using Conventional Culture Methods and Polymerase Chain Reaction Technique

A study was carried out to evaluate three culture media and PCR for the detection of Salmonella spp. to improve Salmonella monitoring program. A total of 109 samples were collected from two farms. Sixty four samples were collected from farm A. These included 16 cloacal swabs collected from broilers before slaughtering, 18 intestinal swabs and 20 caecal swabs collected from broilers after evisceration, and 10 cloacal swabs collected from village chickens. Forty five samples were collected from farm B, which included 15 cloacal swabs from each of village chickens, turkeys, and guinea fowls. Samples were pre-enriched in BPW and investigated by plating them on XLT4 agar after enrichment in selenite cystine broth, BPLS agar after enrichment in Rappaport-Vasilliadis broth, and DIASALM directly after pre-enric hment in BPW. Suspected positive colonies were confirmed biochemically and serologically. DIASALM and BPLS agar were comparatively evaluated against XLT4 agar as the "gold standard" using Kappa statistic to determine the level of agreement between them. A total of 27 (24.77%) Salmonella were detected from the 109 samples. Isolation rates for XLT4, DIASALM, and BPLS were 20.20% (22 out of 109), 17.43% (19 out of 109), and 13.8% (15 out of 109), respectively. The sensitivity and agreement (Kappa statistic) with the "gold standard" for each evaluated detection method were: 70.4% and 0.69 (substantial) for DIASALM and 55.56% and 0.58 (mode rate) for BPLS. For the detection of Salmonella spp. by PCR, bacterial chromosomal DNA was extracted by boiling. Amplicons (429 bp) and (284 bp) derived from primers to the genomic random fragment (primers ST11 and ST15) and invA genes (primers 139 and 141) respectively, were confirmed as Salmonella specific on ethidium bromide-stained agarose gels. Using PCR assay Salmonella was detected 24% (13 out of 54) and 13% (7 out of 54) in broilers in farm A using primers ST11-ST15 and 139-141, respectively. Poultry species in farm B were negative for Salmonella by PCR. A specific primer was used for the detection of Salmonella enteritidis. None of Salmonella detected was Salmonella enteritidis. This study concluded that XLT4 agar is the most sensitive medium and is very specific for the isolation of Salmonella from chicken feces. DIASALM is a good medium for the isolation of Salmonella. The inability of PCR to successfully detect Salmonella specific products from all the samples that were positive for isolation is not clear. However, this would be partly explained by the presence of inhibitor factors in the DNA preparations. In addition, the primer set ST11-ST15 used in this study has not before been tested on cloacal swabs and fecal samples from poultry. Perhaps, with improved DNA extraction method may overcome the inhibitory problem and also low yield of DNA. PCR should be used together with cultivation for the detection of Salmonella especially when the serovar is to be determined

A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Mat

Huyghens, Bohr, Riemann and Galois: Phase-Locking

Several mathematical views of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find them related respectively to the Riemann zeta function and to incomplete Gauss sums.Comment: 18 pages paper written in relation to the ICSSUR'05 conference held in Besancon, France to be published at a special issue of IJMP

On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions