On the second weight of generalized Reed-Muller codes

Not much is known about the weight distribution of the generalized Reed-Muller code RM _q (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m – 1)(q – 1) we then find the first s + 1 – (m – 1)(q–1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.     

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Hyperplane sections of determinantal varieties over finite fields and linear codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated.     

A Generalization of a Theorem of Dehon for Simple t-Designs

A generalization of a theorem of Dehon for simple t-designs is proved. Several new families of 4- and 5-designs having certain additional property regarding their block intersection numbers are obtained by applying this result.The author acknowledges the support of SFB Forschungsschwerpunkt-Geometrie Grant, University of Heidelberg.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Construction of optimal ternary constant weight codes via Bhaskar Rao designs

In this note, we consider a construction for optimal ternary constant weight codes (CWCs) via Bhaskar Rao designs (BRDs). The known existence results for BRDs are employed to generate many new optimal nonlinear ternary CWCs with constant weight 4 and minimum Hamming distance 5.     

The structure of linear codes of constant weight

In this paper we determine completely the structure of linear codes over of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.

     

Designs from subcode supports of linear codes

We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions.      

On the minimum average distance of binary constant weight codes

Let β(n,M,w) denote the minimum average Hamming distance of a binary constant weight code with length n, size M and weight w. In this paper, we study the problem of determining β(n,M,w). Using the methods from coding theory and linear programming, we derive several lower bounds on the average Hamming distance of a binary constant weight code. These lower bounds enable us to determine the exact value for β(n,M,w) in several cases.     

The characterization of binary constant weight codes meeting the bound of Fu and Shen

Fu and Shen gave an upper bound on binary constant weight codes. In this paper, we present a new proof for the bound of Fu and Shen and characterize binary constant weight codes meeting this bound. It is shown that binary constant weight codes meet the bound of Fu and Shen if and only if they are generated from certain symmetric designs and quasi-symmetric designs in combinatorial design theory. In particular, it turns out that the existence of binary codes with even length meeting the Grey–Rankin bound is equivalent to the existence of certain binary constant weight codes meeting the bound of Fu and Shen. Furthermore, some examples are listed to illustrate these results. Finally, we obtain a new upper bound on binary constant weight codes which improves on the bound of Fu and Shen in certain case. This research is supported in part by the DSTA research grant R-394-000-025-422 and the National Natural Science Foundation of China under the Grant 60402031, and the NSFC-GDSF joint fund under the Grant U0675001     

Binary permutation sequences as subsets of Levenshtein codes,spectral null codes,run-length limited codes and constant weight codes

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes. This paper was presented in part at the IEEE Information Theory Workshop, Chengdu, China, October, 2006.     

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

Error detection capability of shortened hamming codes and their dual codes

1. IntroductionIn ant omat ic--rep eat-- request (ARQ ) error- cont rol syst em 3 t he u-ndet ect ed error probability (UEP) of an error-detecting code is one of the most importallt performance characteristics. There are a number of papers dedicated to examining the error detection capabilityfor some well known classes of linear codes, suCh as Reed-Solomon codes, BCH codes andReed-Muller codes. For a general introduction to the theory of error detecting codes, werefer the readers to [if …     

A class of optimal constant composition codes from GDRPs

As a common generalization of constant weight binary codes and permutation codes, constant composition codes (CCCs) have attracted recent interest due to their numerous applications. In this paper, a class of new CCCs are constructed using design-theoretic techniques. The obtained codes are optimal in the sense of their sizes. This result is established, for the most part, by means of a result on generalized doubly resolvable packings which is of combinatorial interest in its own right.      

The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.      

Constructions for generalized Steiner systems <Emphasis Type="Italic">GS</Emphasis>(3, 4, <Emphasis Type="Italic">v</Emphasis>, 2)

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8,10 given by Etzion. In this paper, more small designs are found by computer search and also given are direct constructions based on finite fields and rotational Steiner quadruple systems and recursive constructions using three-wise balanced designs. Some infinite families are also obtained.      

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1