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

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

Quantum Synchronizable Codes From Finite Geometries

Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances

On the rank of incidence matrices in projective Hjelmslev spaces

Let be a finite chain ring with , , and let . Let be an integer sequence satisfying . We consider the incidence matrix of all shape versus all shape subspaces of with . We prove that the rank of over is equal to the number of shape subspaces. This is a partial analog of Kantor's result about the rank of the incidence matrix of all dimensional versus all dimensional subspaces of . We construct an example for shapes and for which the rank of is not maximal

Extracellular ATP drives systemic inflammation, tissue damage and mortality

Systemic inflammatory response syndromes (SIRS) may be caused by both infectious and sterile insults, such as trauma, ischemia-reperfusion or burns. They are characterized by early excessive inflammatory cytokine production and the endogenous release of several toxic and damaging molecules. These are necessary to fight and resolve the cause of SIRS, but often end up progressively damaging cells and tissues, leading to life-threatening multiple organ dysfunction syndrome (MODS). As inflammasome-dependent cytokines such as interleukin-1 beta are critically involved in the development of MODS and death in SIRS, and ATP is an essential activator of inflammasomes in vitro, we decided to analyze the ability of ATP removal to prevent excessive tissue damage and mortality in a murine LPS-induced inflammation model. Our results indeed indicate an important pro-inflammatory role for extracellular ATP. However, the effect of ATP is not restricted to inflammasome activation at all. Removing extracellular ATP with systemic apyrase treatment not only prevented IL-1 beta accumulation but also the production of inflammasome-independent cytokines such as TNF and IL-10. In addition, ATP removal also prevented systemic evidence of cellular disintegration, mitochondrial damage, apoptosis, intestinal barrier disruption and even mortality. Although blocking ATP receptors with the broad-spectrum P2 purinergic receptor antagonist suramin imitated certain beneficial effects of apyrase treatment, it could not prevent morbidity or mortality at all. We conclude that removal of systemic extracellular ATP could be a valuable strategy to dampen systemic inflammatory damage and toxicity in SIRS