Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), and . For , and , we are able to describe small weight codewords geometrically. For , q odd, and for , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]     

A new extension theorem for 3-weight modulo q linear codes over $${\mathbb{F}_ q}$$

We prove that every [n, k, d] _q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .      

Generalized hexagons and Singer geometries

In this paper, we consider a set of lines of with the properties that (1) every plane contains 0, 1 or q + 1 elements of , (2) every solid contains no more than q ² + q + 1 and no less than q + 1 elements of , and (3) every point of is on q + 1 members of , and we show that, whenever (4) q ≠ 2 (respectively, q = 2) and the lines of through some point are contained in a solid (respectively, a plane), then is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in , with q even. We present examples of such sets not satisfying (4) based on a Singer cycle in , for all q.      

On the minimum length of some linear codes

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for when k is odd, for when k is even, and for . This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175). This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 17540129.     

On the Ree Unital

The embedding of a Ree Unital in a finite projective plane Π of order up to q ⁴ is investigated when the Ree group is induced on by a collineation group of Π. In particular, it is shown that such a embedding is not admissible for q ≠ 3, extending in this way a result of Lüneburg dating back to 1965.      

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Elements of Prescribed Order,Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k 1 and be a system of rational functions forming a strongly linearly independent set over a finite field . Let be arbitrarily prescribed elements. We prove that for all sufficiently large extensions , there is an element of prescribed order such that is the relative trace map from onto We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15     

Homogeneous factorisations of Johnson graphs

For a graph Γ, subgroups , and an edge partition of Γ, the pair is a (G, M)-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of , while G permutes the parts of transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples. This paper forms part of an ARC Discovery grant of the last two authors. The second author holds an Australian Research Council Australian Research Fellowship.     

On the nonexistence of certain Hughes generalized quadrangles

In this paper, we prove that the Hermitian quadrangle is the unique generalized quadrangle Γ of order (q ², q ³) containing some subquadrangle of order (q ², q) isomorphic to such that every central elation of the subquadrangle is induced by a collineation of Γ. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

On 2 – (<Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>, 2<Emphasis Type="Italic">n</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>–1) designs with three intersection numbers

The simple incidence structure , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane of order n > 4, is a 2 – (n ²,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer. Supported by M.I.U.R., Università di Palermo.     

Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

The cubic Segre variety in PG(5, 2)

The Segre variety in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U ^# of an r-flat is defined to be

Maximum distance separable poset codes

We derive the Singleton bound for poset codes and define the MDS poset codes as linear codes which attain the Singleton bound. In this paper, we study the basic properties of MDS poset codes. First, we introduce the concept of I-perfect codes and describe the MDS poset codes in terms of I-perfect codes. Next, we study the weight distribution of an MDS poset code and show that the weight distribution of an MDS poset code is completely determined. Finally, we prove the duality theorem which states that a linear code C is an MDS -code if and only if is an MDS -code, where is the dual code of C and is the dual poset of      

On tight projective designs

It is shown that among all tight designs in , where is or , or (quaternions), only 5-designs in (Lyubich, Shatalora Geom Dedicata 86: 169–178, 2001) have irrational angle set. This is the only case of equal ranks of the first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.      

The nonexistence of near-extremal formally self-dual codes

A code is called formally self-dual if and have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over , and . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.      

A new family of maximal curves over a finite field

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n ³ with n = p ^r  > 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n ³(n ³ + 1)(n ² − 1)(n ² − n + 1). Interestingly, has a very large -automorphism group with respect to its genus . Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007.     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.