Classification of Extremal Double Circulant Self-Dual Codes of Lengths 64 to 72

Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.     

On false branch points of incompressible branched immersions

We prove that a branched immersion of a surface with boundary into a differentiable manifold has no false branch points (in fact, no ramified points) if the immersion induces an isomorphism of fundamental groups and some other natural hypotheses are satisfied. This result has immediate applications to Plateau's problem.Work done while the first author was a visiting member of the Max-Planck-Institut für Mathematik at Bonn. Both authors acknowledge the support of Max-Planck-Institut für Mathematik, Bonn and Schwerpunkt Geometrie at Mathematisches Institut, University of Heidelberg     

Mixed-metal cluster chemistry. 28. Core enlargement of tungsten-iridium clusters with alkynyl, ethyndiyl, and butadiyndiyl reagents

Reaction of [WIr3(mu-CO)3(CO)8(eta-C5Me5)] (1c) with [W(C[triple bond]CPh)(CO)3(eta-C5H5)] afforded the edge-bridged tetrahedral cluster [W2Ir3(mu4-eta2-C2Ph)(mu-CO)(CO)9(eta-C5H5)(eta-C5Me5)] (3) and the edge-bridged trigonal-bipyramidal cluster [W3Ir3(mu4-eta2-C2Ph)(mu-eta2-C=CHPh)(Cl)(CO)8(eta-C5Me5)(eta-C5H5)2] (4) in poor to fair yield. Cluster 3 forms by insertion of [W(C[triple bond]CPh)(CO)3(eta-C5H5)] into Ir-Ir and W-Ir bonds, accompanied by a change in coordination mode from a terminally bonded alkynyl to a mu4-eta2 alkynyl ligand. Cluster 4 contains an alkynyl ligand interacting with two iridium atoms and two tungsten atoms in a mu4-eta2 fashion, as well as a vinylidene ligand bridging a W-W bond. Reaction of [WIr3(CO)11(eta-C5H5)] (1a) or 1c with [(eta-C5H5)(CO)2 Ru(C[triple bond]C)Ru(CO)2(eta-C5H5)] afforded [Ru2WIr3(mu5-eta2-C2)(mu-CO)3(CO)7(eta-C5H5)2(eta-C5R5)] [R = H (5a), Me (5c)] in low yield, a structural study of 5a revealing a WIr3 butterfly core capped and spiked by Ru atoms; the diruthenium ethyndiyl precursor has undergone Ru-C scission, with insertion of the C2 unit into a W-Ir bond of the cluster precursor. Reaction of [W2Ir2(CO)10(eta-C5H5)2] with the diruthenium ethyndiyl reagent gave [RuW2Ir2{mu4-eta2-(C2C[triple bond]C)Ru(CO)2(eta-C5H5)}(mu-CO)2(CO)6(eta-C5H5)3] (6) in low yield, a structural study of 6 revealing a butterfly W2Ir2 unit capped by a Ru(eta-C5H5) group resulting from Ru-C scission; the terminal C2 of a new ruthenium-bound butadiyndiyl ligand has been inserted into the W-Ir bond. Reaction between 1a, [WIr3(CO)11(eta-C5H4Me)] (1b), or 1c and [(eta-C5H5)(CO)3W(C[triple bond]CC[triple bond]C)W(CO)3(eta-C5H5)] afforded [W2Ir3{mu4-eta2-(C2C[triple bond]C)W(CO)3(eta-C5H5)}(mu-CO)2(CO)2(eta-C5H5)(eta-C5R5)] [R = H (7a), Me (7c); R5 = H4Me (7b)] in good yield, a structural study of 7c revealing it to be a metallaethynyl analogue of 3.     

Optimal Ternary Linear Rate 1/2 Codes

In this paper, we classify all optimal linear[n, n/2] codes up to length 12. We show that thereis a unique optimal [10, 5, 5] code up to equivalence.     

Improved Bounds for Ternary Linear Codes of Dimension 8 Using Tabu Search

In this paper, new codes of dimension 8 are presented which give improved bounds on the maximum possible minimum distance of ternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Twenty three codes are given which improve or establish the bounds for ternary codes. In addition, a table of upper and lower bounds for d ₃(n, 8) is presented for n 200.     

Lexicodes over rings

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a \(B\) -ordering over these rings and a selection criterion. As examples we give lexicodes over \(\mathbb Z _4\) and \(\mathbb F _2+u\mathbb F _2\) . It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.     

On the Minimum Weight of Codes over F5 Constructed from Certain Conference Matrices

In this paper, codes over F₅ with parameters [36, 18, 12], [48, 24, 15], [60, 30, 18], [64, 32, 18] and [76, 38, 21] which improve the previously known bounds on the minimum weight for linear codes over F₅ are constructed from conference matrices. Through shortening and truncating, the above codes give numerous new codes over F₅ which improve the previously known bounds on minimum weights.