Let F n q be a vector space of dimension n over the finite field F q . A q-analog of a Steiner system (also known as a q-Steiner system), denoted S q (t,k,n), is a set S of k-dimensional subspaces of F n q such that each t-dimensional subspace of F n q is contained in exactly one element of S. Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k = n. In this paper, the first nontrivial q-Steiner systems with t 2 are constructed. Specifically, several nonisomorphic q-Steiner systems S 2 (2, 3, 13) are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions.
In the present paper we consider a q-analog of t − (v, k, λ)-designs. It is canonic since it arises by replacing sets by vector spaces over GF (q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom. Dedicata vol. 63, pp. 247-253 (1996)] they are called t − (v, k, λ; q)-designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package.In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t − (v, k, λ)-designs on sets, obtaining this way new results on the existence of t − (v, k, λ; q)-designs on spaces for further quintuples (t, v, k, λ; q) of parameters. We present several 2 − (6, 3, λ; 2)-designs, 2 − (7, 3, λ; 2)-designs and, as far as we know, the very first 3-designs over GF (q).
In network coding a constant dimension code consists of a set of k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of F_q^n. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper we give a classification of cyclic orbit codes and propose a decoding procedure for a particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor
•The contributions of genetic and environmental factors to differential reproductive success across hybrid zones have rarely been tested. Here, we report a manipulative experiment that simultaneously tested endogenous (genetic-based) and exogenous (environmental-based) selection within a hybrid zone. We transplanted mated pairs of two chickadee species {Poecile atricapilla and P. carolinensis) and their hybrids into isolated woodlots within their hybrid zone and monitored their reproductive success. Although clutch sizes were similar, based on an estimate of the genetic compatibility of a pair, hybrid pairs produced fewer nestlings and fledglings than did pairs of either parental species. According to a linear model generated from the data, a pure pair of either parental species would be expected to produce 1.91-2.48 times more fledglings per nesting attempt, respectively, than the average or least compatible pair in the experiment. Our result of decreased reproduction for hybrid pairs relative to parental species pairs within same environment (the hybrid zone in this experiment) support the endogenous selection hypothesis for maintenance of this hybrid zone. Because the experiment was conducted entirely within the hybrid zone (i.e., the same environment for parental and hybrid pairings), our data do not support the exogenous selection hypothesis as it predicts either all pairings doing poorly or the hybrid pairs more successful than the parental pairs.
The projective space of order n over the finite field F q , denoted here as P q (n), is the set of all subspaces of the vector space F n q . The pro-size M such that the distance between any two codewords (subspaces) is at least d. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for errorcorrection in networks: an (n, M, d) code can correct t packet errors and ρ packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2ρ < d. This motivates our interest in such codes.In this paper, we examine the two fundamental concepts of "complements" and "linear codes" in the context of P q (n). These turn out to be considerably more involved than their classical counterparts. These concepts are examined from two different points of view: coding theory and lattice theory. Our results reveal a number of surprising phenomena pertaining to complements and linearity in P q (n) and gives rise to several interesting problems.
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