Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

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 .      

On self-dual codes over {\mathbb{F}}_5

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths [22, 26, 28, 32–40]. In particular, we prove that there is no [22, 11, 9] self-dual code over , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.      

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

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      

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.      

On $${\mathcal{F}}$$ -supplemented subgroups of finite groups

Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H _G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

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 .      

Efficient <Emphasis Type="Italic">p</Emphasis>th root computations in finite fields of characteristic <Emphasis Type="Italic">p</Emphasis>

We present a method for computing pth roots using a polynomial basis over finite fields of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension of our method requires p − 1 scalar multiplications of elements in by elements in . In addition, our method requires at most additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z ^1/p , defined as the Hamming weight of z ^1/p or , is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260–270, 2007) give is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find always.      

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

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.      

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.      

Self-dual codes over \mathbb{Z}_8 and \mathbb{Z}_9

We study self-dual codes over the rings and . We define various weights and weight enumerators over these rings and describe the groups of invariants for each weight enumerator over the rings. We examine the torsion codes over these rings to describe the structure of self-dual codes. Finally we classify self-dual codes of small lengths over .     

On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.     

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]     

Fq-linear Cyclic Codes over $$ F{_q^m}$$ : D FT Approach

Codes over that are closed under addition, and multiplication with elements from F_q are called F_q-linear codes over . For m 1, this class of codes is a subclass of nonlinear codes. Among F_q-linear codes, we consider only cyclic codes and call them F_q-linear cyclic codes (F_q LC codes) over The class of F_q LC codes includes as special cases (i) group cyclic codes over elementary abelian groups (q=p, a prime), (ii) subspace subcodes of Reed–Solomon codes (n=q^m–1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over F_q (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular F_q-basis of , any F_qLC code over can be viewed as an m-quasi-cyclic code of length mn over F_q. In this correspondence, we obtain transform domain characterization of F_q LC codes, using Discrete Fourier Transform (DFT) over an extension field of The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over . We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual F_q LC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.AMS classification 94B05     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S ₀. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S ₀. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S ₀ attached to each other by narrow necks and winding a large number of times around before closing up on themselves; and are topologically equivalent to .