Weight Enumerators of Double Circulant Codes and New Extremal Self-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.     

New Optimal Self-Dual Codes over GF(7)

In this paper, double circulant self-dual codes over GF(7) are presented, including [12,6,6] codes which are new optimal codes. It is shown that the supports of the codewords of weights 9, 10 and 11 in double circulant [20,10,9] codes form 3-designs. For larger lengths, some good self-dual codes are constructed from weighing matrices. Received: June 24, 1996 / Revised: February 28, 1997     

Higher Weights for Ternary and Quaternary Self-Dual Codes*

We study higher weights applied to ternary and quaternary self-dual codes. We give lower bounds on the second higher weight and compute the second higher weights for optimal codes of length less than 24. We relate the joint weight enumerator with the higher weight enumerator and use this relationship to produce Gleason theorems. Graded rings of the higher weight enumerators are also determined. This work was supported in part by Northern Advancement Center for Science & Technology and the Natural Sciences and Engineering Research Council of Canada.     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

Mixed-metal cluster chemistry. 29. Core expansion and ligand-driven metal exchange at group 6-iridium clusters

Reactions of the tetrahedral clusters MoIr3(mu-CO)3(CO)8(eta-L) (L = C5HMe4, C5Me5) with the carbonylmetalate anions [Mo(CO)3(eta-L)]- afford the trigonal bipyramidal clusters Mo2Ir3(mu3-H)(mu-CO)2(CO)9(eta-L)2 (L = C5HMe4 (3c), 74%; L = C5Me5 (3d), 55%) in which the group 6 metal atoms occupy the apexes; reaction of the cyclopentadienylmolybdenum-containing analogues or their cyclopentadienyltungsten-containing homologues failed to afford analogous products. Reactions of MIr3(mu-CO)3(CO)8(eta-C5H5) (M = Mo, W) with [M(CO)3(eta-L)]- (L = C5HMe4, C5Me5) afford the core-expanded heteroapex clusters M2Ir3(mu3-H)(mu-CO)2(CO)9(eta-C5H5)(eta-L) (M = Mo, L = C5HMe4 (5c), 9%, L = C5Me5 (5d), 4%; M = W, L = C5Me5 (6d), 5%) in low yield, together with the homoapex clusters M2Ir3(mu3-H)(mu-CO)2(CO)9(eta-L)2 (M = Mo, L = C5HMe4 (3c), 81%, L = C5Me5 (3d), 60%; M = W, L = C5Me5 (4d), 5%) in much higher yield for the Mo-containing examples. The identities of clusters 3c,d, 4d, and 5c,d have been confirmed by single-crystal X-ray diffraction studies, with the same disposition of ligands about the trigonal bipyramidal cluster cores being observed in each case, a ligand arrangement that has been examined by complementary density functional theory studies. While cluster 5d is accessible as above, no reaction is observed from MoIr3(mu-CO)3(CO)8(eta-C5Me5) and [M(CO)3(eta-C5H5)]-. Treating MoIr3(mu-CO)3(CO)8(eta-C5H5) with 1 equiv of [M(CO)3(eta-C5Me5)]- affords 5d as the major product, a further 1 equiv affording some MoIr3(mu-CO)3(CO)8(eta-C5Me5) and a third 1 equiv giving a good yield of 3d. This is consistent with reaction proceeding by apex fragment addition, followed by apex fragment elimination, and finally a further apex fragment addition, the homometallic incoming apexes being distinguished from the departing vertices by their highly methylated cyclopentadienyl ligands. Spectroscopic data suggest that the electron density at these disparate-metal-containing cluster cores is tunable by progressive (conceptual) cyclopentadienyl alkylation.     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

MDS and self-dual codes over rings

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings. We also define the Reed–Solomon codes over principal ideal rings.