The equivalence of certain equidistant binary codes and symmetric bibds

We study equidistant codes of length 4k + 1 having (constant) weight 2k, and (constant) distance 2k between codewords. The maximum number of codewords is 4k; this can be attained if and only ifk = (u ² +u)/2 (for some integeru) and there exists a ((2u ² + 2u + 1,u ², (u ² −u)/2) — SBIBD. Also, one can construct such a code, with 4k − 1 codewords, from a (4k − 1, 2k − 1,k − 1) — SBIBD. Supported, in part by NSERC grants U0217 (D. R. Stinson), A3558 (G. H. J. van Rees).     

Nonexistence of certain pseudogeometric graphs

Izo($r$) is the pseudogeometric graph for pG${}_r(s,t)$ which satisfies the Krein equality and of which the local graph also satisfies the Krein equality. We prove that Izo($r$) exists if and only if a spherical tight 4-design on $S^{{(2r+1)}^2?4}$ exists. Nonexistence of infinitely many spherical tight 4-designs is well known. It gives the nonexistence of Izo($r$) for infinitely many $r$. Especially, Izo(4) does not exist, which was the first open case.     

Nonexistence of invariant semigroups in affine symmetric spaces

Let be an affine symmetric space with a simple Lie group, an involutive automorphism of and an open subgroup of the -fixed point group . It is proved here that the existence of a proper semigroup with and implies that is of Hermitian type, as conjectured by Hilgert and Neeb [4]. When exists, it turns out that it leaves invariant an open -orbit in a minimal flag manifold of . A byproduct of our approach is an alternate proof of the maximality of the compression semigroup of an open orbit (see Hilgert and Neeb [31]). Received: 10 September 1999 / Published online: 23 July 2001     

Nonexistence of some quantum jump codes with specified parameters

Quantum jump codes (JC) are quantum codes which correct errors caused by quantum jumps. A spontaneous emission error design (SEED) has been introduced to construct quantum JC. In this paper the nonexistence of some \((n,m,t)_k\) -JCs is proved, where \(m={n-t\atopwithdelims ()k-t},\, k\ge t+1,\, \text{ and }\, (n,t,k)=(2k+1,1,k), (2k,2,k),(7,2,3),(8,3,4)\) . It is also shown that, for these parameters with \(t=1,2\) , the existence of an \((n,m,t)_k\) -JC is equivalent to that of a \(t-(n,k;m)\) -SEED.     

Nonexistence of certain planes of order 10 and 12

It is shown here that no plane of order 12 exists which has a group of elations of order 12. It is also shown that no plane of order 10 exists which has a multiplicative loop that is a group.     

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R ³ homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane. Received: November 26, 1998     

Nonexistence of Brownian models for certain multiclass queueing networks

We present two multiclass queueing networks where the Brownian models proposed by Harrison and Nguyen [3,4] do not exist. If self-feedback is allowed, we can construct such an example with a two-station network. For a three-station network, we can construct such an example without self-feedback.Research supported in part by Texas Instruments Corporation Grant 90456-034.     

Extremal self-dual codes from symmetric designs

A method is given for constructing extremal binary self-dual [42, 21, 8]-codes from certain symmetric 2-(41, 16, 6) designs.     

Nonexistence of some linear codes over the field of order four

We consider the problem of determining $n_4(5,d)$, the smallest possible length $n$ for which an ${[n,5,d]}_4$ code of minimum distance $d$ over the field of order 4 exists. We prove the nonexistence of ${[g_4(5,d)+1,5,d]}_4$ codes for $d=31,47,48,59,60,61,62$ and the nonexistence of a ${[g_4(5,d),5,d]}_4$ code for $d=138$ using the geometric method through projective geometries, where $g_q(k,d)={\sum }_{i=0}^{k?1}?d∕q^i?$. This yields to determine the exact values of $n_4(5,d)$ for these values of $d$. We also give the updated table for $n_4(5,d)$ for all $d$ except some known cases.     

Distances between certain symmetric spaces

This paper is devoted to estimates of Banach-Mazur distances between finite-dimensional symmetric spaces. Lower and upper estimates are obtained in terms of fundamental sequences of symmetric spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 218–224, 1981.     

Few-cosine spherical codes and Barnes-Wall lattices

Using Barnes-Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape ²1+8⋅GL(4,2). The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters (16,256,6), a Nordstrom-Robinson code, and gives a context for determining its automorphism group, which has form .     

Nonexistence of axially symmetric,stationary solution of Einstein Vacuum Equation with disconnected symmetric event horizon

In this short note, we prove that if an axially symmetric, asymptotically flat, stationary solution of the Einstein Vacuum Equation admits an involution interchanging the components of the event horizons, then the force between at least two of the event horizons must be positive. The proof is based on an observation that in this case the force can be expressed as a line integral with positive integrand.     

Inversion of certain symmetric band matrices

A procedure is given for inversion of a symmetric band matrix with all elements in a certain diagonal equal. Starting from the main diagonal we have the elementsk,k–1,k–2, ... 2,1 with zeros in the remaining diagonals. Letting the second order difference operator ² operate on the rows of the matrix we obtain a new matrix which can be inverted by a partition method after certain permutations of the elements.     

Weakly symmetric spaces and spherical varieties

Weakly symmetric homogeneous spaces were introduced by A. Selberg in 1956. We prove that, for a real reductive algebraic group, they can be characterized as the spaces of real points of affine spherical homogeneous varieties of the complexified group. As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.Supported by the Alexander von Humboldt Foundation and Russian Foundation for Basic Research, Grant No. 95-01-01263.Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, Russian Foundation for Basic Research, Grant No. 98-01-00598, and the Alexander von Humboldt Foundation.     

Properties of certain symmetric stable distributions

Necessary and sufficient conditions are presented for jointly symmetric stable random vectors to be independent and for a regression involving symmetric stable random variables to be linear. The notion of n-fold dependence is introduced for symmetric stable random variables, and under this condition we determine all monomials in such random variables for which moments exist.