LDPC codes associated with linear representations of geometries

We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K

A study of (xvt,xvt−1)-minihypers in PG(t,q)

AbstractWe study (xvt,xvt−1)-minihypers in PG(t,q), i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case t=2. We establish a new link with coding theory and we use this link to construct several new infinite classes of (xvt,xvt−1)-minihypers in PG(t,q) that cannot be written as an integer sum of hyperplanes

On the dual code of points and generators on the Hermitian variety H(2n+1,q²)

We study the dual linear code of points and generators on a non-singular Hermitian variety H(2n + 1, q(2)). We improve the earlier results for n = 2, we solve the minimum distance problem for general n, we classify the n smallest types of code words and we characterize the small weight code words as being a linear combination of these n types

Large weight code words in projective space codes

AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points

On KM-arcs in small Desarguesian planes

In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KMq,t-arcs for q <= 32, up to projective equivalence. We also construct a KM64,4-arc; as t=4 was the only value for which the existence of a KM64,t-arc was unknown, this fully settles the existence problem for q <= 64

Juvenile Hippocampus guttulatus from a neuston tow at the French-Belgian border

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Codes of Desarguesian projective planes of even order, projective triads and (q+t,t)-arcs of type (0,2,t)

AbstractWe study the binary dual codes associated with Desarguesian projective planes PG(2,q), with q=2h, and their links with (q+t,t)-arcs of type (0,2,t), by considering the elements of Fq as binary h-tuples. Using a correspondence between (q+t,t)-arcs of type (0,2,t) and projective triads in PG(2,q), q even, we present an alternative proof of the classification result on projective triads. We construct a new infinite family of (q+t,t)-arcs of type (0,2,t) with t=q4, using a particular form of the primitive polynomial of the field Fq

Floating seaweed as a vector for travelling organisms

The neuston, i.e. the fauna inhabiting the upper layer of oceans and seas, is strongly influenced by the occurrence of floating patches composed of detached coastal seaweed fragments. Ephemeral floating seaweeds harbour a diverse fauna originating from attached seaweeds, the strandline of beaches, the surrounding and underlying water column, the seafloor or the air. These organisms colonise the seaweeds for various reasons, usually including the provision of shelter, food or attachment substrate. The association behaviour of these organisms and their use of the resources offered by floating seaweeds potentially have important ecological consequences, such as the possibility of passive dispersal of associated fauna to new, distant locations by means of rafting. During this study, different aspects of raft-associated ecology were addressed. The results demonstrate that the habitat formed by floating seaweeds is very complex. Although the presence of floating seaweeds in the neuston can, to a certain degree, be seasonally predicted (storms, seasonal release of fertile structures), the habitat that they form is still very patchy and unstable. Consequently, most species found in association with ephemeral floating seaweed patches are opportunistic of nature. The association behaviour of the encountered species and their (optimal) use of the transient resources offered by floating seaweeds can, in certain circumstances, result in the passive dispersal of associated fauna to new, and distant locations by means of rafting. The process of rafting strongly depends on the longevity of the seaweed raft, which is in turn significantly influenced by temperature and grazing pressure. In favorable conditions, seaweed rafts can potentially cover great distances, carrying with them rafting fauna that are able to survive a long journey in the neuston