Codes with the Same Weight Distributions as the Goethals Codes and the Delsarte-Goethals Codes

The Goethals code is a binary nonlinear code of length 2^m+1 which has codewords and minimum Hamming distance 8 for any odd . Recently, Hammons et. al. showed that codes with the same weight distribution can be obtained via the Gray map from a linear code over Z ₄of length 2^m and Lee distance 8. The Gray map of the dual of the corresponding Z ₄ code is a Delsarte-Goethals code. We construct codes over Z ₄ such that their Gray maps lead to codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes.     

On the Existence of Large Sets of <Emphasis Type="Italic">t</Emphasis>-designs of Prime Sizes

In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.AMS Classification: 05B05Communicated by:L. TeirlinckThis research was in part supported by a grant from IPM (No. 83050312).     

完美门限方案的组合构造

部分平衡t-设计t-(v; b;w; 1; 0) (X;A) 称为可划分的, 如果它同时也是一个部分平衡(t-1)-设计(t -1)-(v; b;w; λ_t-1; 0) 并且可将区组集A划分为A₁;…;A_λt-1; 使得每个(X;A_i) (1≤i≤λ_t-1)是一个部分平衡(t-1)-设计(t-1)-(v; b/λ_t-1;w; 1; 0). 本文证明可划分部分平衡t-设计PPBD t-(v; b;w;λ_t-1; 1; 0) 的存在性蕴含着完美(t;w; v;λ_t-1)-门限方案的存在性; 而且在某些情况下, 最优可划分部分平衡t-设计OPPBD(t;w; v) 的存在性等价于最优(t;w; v)-门限方案的存在性. 由此我们得到了最优(t;w; v)-门限方案的一些新的无穷类.     

A Steiner 5-Design on 36 Points

Up to now, all known Steiner 5-designs are on q + 1 points where q 3 (mod 4) is a prime power and the design is admitting PSL(2, q) as a group of automorphisms. In this article we present a 5-(36,6,1) design admitting PGL(2, 17) × C ₂ as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2, 17) × Id ₂ of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits.     

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.      

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.     

Ranks of q-Ary 1-Perfect Codes

The rank of a q-ary code C of length n, r(C), isthe dimension of the subspace spanned by C. We establish the existence of q-ary 1-perfectcodes of length for m 4 and r(C)= n – m + s for each s {1,,m}. This is a generalization of the binary case proved by Etzion and Vardy in[4].     

Bounds for Covering Codes over Large Alphabets

Let K_q(n,R) denote the minimum number of codewords in any q-ary code of length n and covering radius R. We collect lower and upper bounds for K_q(n,R) where 6 ≤ q ≤ 21 and R ≤ 3. For q ≤ 10, we consider lengths n ≤ 10, and for q ≥ 11, we consider n ≤ 8. This extends earlier results, which have been tabulated for 2 ≤ q ≤ 5. We survey known bounds and obtain some new results as well, also for s-surjective codes, which are closely related to covering codes and utilized in some of the constructions.AMS Classification: 94B75, 94B25, 94B65Gerzson Kéri - Supported in part by the Hungarian National Research Fund, Grant No. OTKA-T029572.Patric R. J. Östergård - Supported in part by the Academy of Finland, Grants No. 100500 and No. 202315.     

All q m -ary Cyclic Codes with Cyclic q-ary Image Are Known

We give a simpler presentation of recent work byLeonard, Séguin and Tang-Soh-Gunawan. Our methodsimply as a new result that even in the repeated-root casethere are no more q ^m -arycyclic codes with cyclic q-ary image than thosegiven by Séguin.     

Compact corigid objects in triangulated categories and co-<Emphasis Type="Italic">t</Emphasis>-structures

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, , of a triangulated category, , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on whose heart is equivalent to Mod(End()^op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, , of a triangulated category, , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End()^op), and hence an abelian subcategory of .      

Large sets of <Emphasis Type="Italic">t</Emphasis>-designs from groups

This paper addresses questions related to the existence and construction of large sets of t-(v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa’s 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of “highlight” topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them. (Communicated by Peter Horák)     

Generalized <Emphasis Type="Italic">SV</Emphasis>-modules

Given an arbitrary quasiprojective right R-module P, we prove that every module in the category σ(P) is weakly regular if and only if every module in σ(M/I(M)) is lifting, where M is a generating object in σ(P). In particular, we describe the rings over which every right module is weakly regular.     

Completely Regular Designs of Strength One

We study a class of highly regular t-designs. These are the subsets of vertices of the Johnson graph which are completely regular in the sense of Delsarte [2]. In [9], Meyerowitz classified the completely regular designs having strength zero. In this paper, we determine the completely regular designs having strength one and minimum distance at least two. The approach taken here utilizes the incidence matrix of (t+1)-sets versus k-sets and is related to the representation theory of distance-regular graphs [1, 5].     

On sylowizers in finite <Emphasis Type="Italic">p</Emphasis>-soluble groups

In general, given a finite group G, a prime p and a p-subgroup R of G, the sylowizers of R in G are not conjugate. In this paper we afford some conditions to achieve the conjugation of the sylowizers of R in a p-soluble group G, among others

Estimation Methods for the Multivariate <Emphasis Type="Italic">t</Emphasis> Distribution

The known estimation and simulation methods for multivariate t distributions are reviewed. A review of selected applications is also provided. We believe that this review will serve as an important reference and encourage further research activities in the area.     

Unilateral <Emphasis Type="Italic">o</Emphasis>-Groups

For a big number of varieties of groups close to Engelian, it is proved that a variety of lattice-ordered groups generated by all linearly ordered groups in the class does not coincide with the variety of all o-approximable lattice-ordered groups. Supported by FP “Universities of Russia” grant No. UR.04.01.001. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 20–27, January–February, 2006.     

Designs and Representation of the Symmetric Group

We study (generalized) designs supported by words of given composition. We characterize them in terms of orthogonality relations with Specht modules; we define some zonal functions for the symmetric group and we give a closed formula for them, indexed on ordered pair of semi-standard generalized tableaux: Hahn polynomials are a particular case. We derive an algorithm to test if a set is a design. We use it to search designs in some ternary self-dual codes.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

<Emphasis Type="Italic">q</Emphasis>-Selberg Integrals and Macdonald Polynomials

Using the theory of Macdonald polynomials, a number of q-integrals of Selberg type are proved. 2000 Mathematics Subject Classification: Primary—33D05, 33D52, 33D60