Howard Rumsey

New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities

With the Delsarte-MacWilliams inequalities as a starting point, an upper bound is obtained on the rate of a binary code as a function of its minimum distance. This upper bound is asymptotically less than Levenshteins bound, and so also Eliass.

Alternating sign matrices and descending plane partitions

Abstract An alternating sign matrix is a square matrix such that (i) all entries are 1, −1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math. 53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math. 66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.

Determinants and alternating sign matrices

Let M be an n by n matrix. By a connected minor of M of size k we mean a minor formed from k consecutive rows and k consecutive columns. We give formulas for det M in terms of connected minors, one involving minors of two consecutive sizes and one involving minors of three consecutive sizes. The formulas express det M as sums indexed by sets of alternating sign matrices. These matrices are described here and by W. H. Mills, D. P. Robbins, and H. Rumsey, Jr. (Invent. Math.66 (1982), 73–87; J. Combin. Theory Ser. A.34 (1983), 340–359). The former study has led to the solution of Macdonalds conjecture on cyclically symmetric plane partitions (G. E. Andrews, Invent. Math.53 (1979), 193–225; I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” p. 53, Oxford Univ. Press (Clarendon), Oxford, 1979).

Euler products, cyclotomy, and coding☆

Abstract We present a familiar manipulation of Euler products and L -series, and derive a recent theorem of Carlitz and an old theorem of Davenport-Hasse. We then show that the theorem of Davenport-Hasse can be used to shed considerable light on certain questions about “irreducible cyclic codes,” and the cyclotomy of finite fields, including some recent work of the Lehmers.

Self-complementary totally symmetric plane partitions

Abstract A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box X n = [1, n ] × [1, n ] × [1, n ] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box X n that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex ( n + 1, n + 1, n + 1). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless n = 2 m is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m . We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.

Enumeration of a symmetry class of plane partitions

Abstract A cyclically symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box n =[1, n ]×[1, n ]×[1, n ] and which is mapped to itself by cyclic permutations of the coordinate axes. Given a cyclically symmetric plane partition with Ferrers graph F , we can form its transpose-complement, the plane partition whose Ferrers graph is the set of all triples ( i , j , k ) such that ( n + 1 − j , n + 1 − i , n + 1 − k ) ∋ F . This is again a cyclically symmetric plane partition. A cyclically symmetric plane partition is tc-symmetric if it is equal to its transpose-complement. THis cannot occur unless n is even. In this paper we show that the number of tc -symmetric plane partitions in H 2n is given by π k=0 n−1 (3K+1)(6k)!(2k)! (4k+1)!(4k)! We show that, with a suitable assignment of weights, the generating function for tc -symmetric plane partitions divides the generating function for all cyclically symmetric plane partitions. We give analogous results for other classes of plane partitions including descending plane partitions.

A low-rate improvement on the Elias bound (Corresp.)

An upper bound on the minimum distance of binary blocks codes, which is superior to Elias bound for R < 0.0509^+, is obtained. The new bound has the same derivative(-infty) at R = 0 as Gilberts lower bound. (Elias bound has derivative-ln 2 at R = 0).

Enumeration of subspaces by dimension sequence

Abstract Let F be the finite field with q elements and let V be an m-dimensional vector space over F. Fix a linear endomorphism A of V. Suppose that X ⊆ V is a subspace. Let X(0) = 0, X(1) = X, and X(k) = X(k − 1) + Ak − 1(X) for k > 1. For k ⩾ 1 let jk be the dimension of the quotient space X (k) X (k−1) . The m-tuple j = (j1, j2, …, jm) is called the dimension sequence of the subspace X. In this paper we consider the problem of determining the numbers C(j) of subspaces of V which have dimension sequence j. We derive two surprisingly simple formulas. First, if A is a shift operator (nilpotent with 1-dimensional null space) then C(j)= ∏ k=2 m q j 2 k j k−1 j k . Second, if A is simple (no non-trivial invariant F-subspaces) then C(j)= q m −1 q j1 −1 ∏ k=2 m q jk(jk−1) j k−1 j k . We known of no simple counting proof of these formulas. Our derivation finds the C(j)s as the solution to a set of linear equations obtained with Mobius inversion on the lattice of subspaces of V.

Terrestrial Quadstatic Interferometric Radar Observations of Mars

A new technique for resolving the ambiguity inherent in delay-Doppler radar observations of Mars has been developed and implemented using a suite of data collected during 2001, 2003, and 2005 oppositions. New recording systems, processing techniques, and, most importantly, the addition of a fourth receiving telescope allow for the high-resolution mapping of Mars radar properties. In this paper, we develop a maximum likelihood method to probabilistically estimate the contributions to the received radar signal from the ambiguous resolution cells. Our delay-Doppler interferometric radar observations are designed to map the radar properties of Mars surface while disregarding the historically typical goal of measuring topography, instead using Mars topography as a priori knowledge. Example data from the September 27, 2003 observation over Mars southern highlands, including Maadim Vallis and Gusev Crater, and the June 7, 2001 observation crossing Terra Meridiani are presented to demonstrate the effectiveness of the technique. Analysis of these observations predicted a root-mean-square (rms) roughness, or slopes, for the Mars Exploration Rover (MER) Spirit landing site of 1.80° ± 0.75°. Similarly, analysis of the Gusev Crater landing site of MER Opportunity predicted rms slopes of 1.1° ± 0.1°. Both predictions were validated by analysis of in situ rover images.