LDPC codes from the Hermitian curve

In this paper, we study the code C which has as parity check matrix H the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in ( Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that C has a double cyclic structure and that by shortening in a suitable way H it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix H via a geometric approach

On some subvarieties of the Grassmann variety

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $\Pi$. It is known that the Pl\"{u}cker embedding of the elements of $\mathcal S$ is a variety of $PG(r^t-1,q)$, say ${\mathcal V}_{rt}$. In this paper, we describe the image under the Pl\"{u}cker embedding of the elements of $\Lambda$ and we show that it is an $m$-dimensional algebraic variety, projection of a Veronese variety of dimension $m$ and degree $t$, and it is a suitable linear section of ${\mathcal V}_{rt}$.Comment: Keywords: Grassmannian, linear set, Desarguesian spread, Schubert variet

On symplectic semifield spreads of PG(5,q2), q odd

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ( 5 , q2), for q2> 2 .38odd, whose associated semifield has center containing Fq. Equivalently, we classify, up to isotopy, commutative semifields of order q6, for q2> 2 .38odd, with middle nucleus containing q2Fq2and center containing q Fq

On varieties defined by large sets of quadrics and their application to error-correcting codes

Let $U$ be a $({ k-1 \choose 2}-1)$-dimensional subspace of quadratic forms defined on $\mathrm{PG}(k-1,{\mathbb F})$ with the property that $U$ does not contain any reducible quadratic form. Let $V(U)$ be the points of $\mathrm{PG}(k-1,{\mathbb F})$ which are zeros of all quadratic forms in $U$. We will prove that if there is a group $G$ which fixes $U$ and no line of $\mathrm{PG}(k-1,{\mathbb F})$ and $V(U)$ spans $\mathrm{PG}(k-1,{\mathbb F})$ then any hyperplane of $\mathrm{PG}(k-1,{\mathbb F})$ is incident with at most $k$ points of $V(U)$. If ${\mathbb F}$ is a finite field then the linear code generated by the matrix whose columns are the points of $V(U)$ is a $k$-dimensional linear code of length $|V(U)|$ and minimum distance at least $|V(U)|-k$. A linear code with these parameters is an MDS code or an almost MDS code. We will construct examples of such subspaces $U$ and groups $G$, which include the normal rational curve, the elliptic curve, Glynn's arc from \cite{Glynn1986} and other examples found by computer search. We conjecture that the projection of $V(U)$ from any $k-4$ points is contained in the intersection of two quadrics, the common zeros of two linearly independent quadratic forms. This would be a strengthening of a classical theorem of Fano, which itself is an extension of a theorem of Castelnuovo, for which we include a proof using only linear algebra

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

Non-intersecting Ryser hypergraphs

A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szab\'{o} that all $3$-Ryser hypergraphs with matching number $\nu > 1$ are essentially obtained by taking $\nu$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for $r = 4$ by giving a computer generated example of a $4$-Ryser hypergraph with $\nu = 2$ whose vertex set cannot be partitioned into two sets such that we have an intersecting $4$-Ryser hypergraph on each of these parts. Here we construct new infinite families of $r$-Ryser hypergraphs, for any given matching number $\nu > 1$, that do not contain two vertex disjoint intersecting $r$-Ryser subhypergraphs.Comment: 8 pages, some corrections in the proof of Lemma 3.6, added more explanation in the appendix, and other minor change

A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results

On the algebraic variety Vr,t

AbstractThe variety Vr,t is the image under the Grassmannian map of the (t−1)-subspaces of PG(rt−1,q) of the elements of a Desarguesian spread. We investigate some properties of this variety, with particular attention to the case r=2: in this case we prove that every t+1 points of the variety are in general position and we give a new interpretation of linear sets of PG(1,qt)

Development of procedures to perform nanoindentation tests on different bone structures

Negli ultimi anni, la nanoindentazione è emersa come potente tecnica per indagare le proprietà micromeccaniche dell'osso. L'indentazione consiste nel premere una punta rigida con una forza nota in un semispazio semi-infinito e nel misurare direttamente o indirettamente l'area di contatto. L'obiettivo principale di questo lavoro è stato quello di sviluppare una procedura per eseguire test di nanoindentazione al fine di studiare le proprietà elastiche e inelastiche di diverse strutture ossee. Dalle misure di nanoindentazione sono stati ricavati i valori di reduced modulus, hardness, indentation modulus ed elastic modulus. L'idea era di eseguire test di nanoindentazione sia per applicazioni precliniche che cliniche e per questo motivo, i tests sono stati effettuati sia su ossa di topo che su ossa umane affette da una particolare condizione patologica, chiamata Osteogenesi Imperfetta. È la prima volta che questi tests vengono eseguiti su tibie di topo, nello specifico su fette di quattro tibie di due ceppi (C57B1/6 e Balb/C), sia su osso corticale che trabecolare. Abbiamo trovato che il modulo elastico varia tra 16.50 ± 7.10 GPa (C57B1/6, osso trabecolare) e 25.08 ± 5.21 GPa (Balb/C, osso corticale). L’hardness varia tra 0.62 ± 0.27 GPa (C57B1/6, osso trabecolare) e 0.96 ± 0.20 GPa (Balb/C, osso corticale). Le nanoindentazioni sul campione di OI (proveniente dall’arto superiore) sono state condotte su diverse fette, per analizzare le potenziali differenze tra le due regioni e le quattro sezioni. Abbiamo trovato un modulo elastico di 12.14 ± 5.79 GPa e l’hardness di 0.49 ± 0.21 GPa. In conclusione, abbiamo sviluppato questo nuovo protocollo che può essere applicato a diversi lavori futuri. Ad esempio, per le applicazioni precliniche aumentando il numero di topi diversi o per applicazioni cliniche aumentando il numero dei campioni di OI, raccogliendo campioni con diversi tipi di OI, indagando l'effetto dei trattamenti o confrontando le ossa di OI con osso sano