Greg Coxson

• United States Naval Academy

Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist code pairs for which the mechanism behind shared autocorrelation is not so simple; these are called shared-autocorrelation (or shared-ACS) binary code pairs. They are useful for, among other things, constructing b...

Binary Golay pairs are examined with an interest in finding examples in which members of the pair can be replaced with other codes sharing the same autocorrelation, but not due to negation and reversal. These are referred to as shared-autocorrelation (or shared- ACS) Golay pairs. Golay pairs can always formed from other Golay pairs by reversing and...

The quad-phase codes of a given length may be organized into equivalence classes relative to a set of operations preserving autocorrelation peak sidelobe level. Knowledge about these equivalence classes can be exploited for faster searches or for efficient listing of low-PSL codes. This paper is part of an effort to enumerate autocorrelation peak s...

Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist “shared-autocorrelation” code pairs for which the mechanism behind shared autocorrelation is not so simple. In this paper, exhaustive search is used to find representatives for all such code pairs for lengths 2 to...

Bat biosonar offers a natural source for biomimetic design of radar waveforms with inspirations falling into two categories: (1) biosonar principles similar to ones already employed in radar and (2) principles used by bats that operate in ways not yet understood or not yet embraced yet for radar. This second type offers the possibility for driving...

A radar signal processor subchain containing a pulse compression module employing binary $\pm1$ phase codes is studied in an effort to understand the behavior of the output mean and variance as functions of phase code imbalance, in noise-limited environments. It is a relatively simple subchain consisting of an R/Theta limiter, a pulse compressor u...

The Coffee-Can radar was designed by Professor Gregory Charvat as a simple low-cost radar system for small teams of students to build and test during an intersession course at MIT. The name derives from the use of coffee cans as transmit and receive antennas. Since the success of Charvat’s course, the coffee-can radar has been used at a number of s...

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “PSL-preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of 2 x N binary grids with r...

A complementary set of K binary ±1 codes of length N has the useful property that the sum of the autocorrelation sequences of the codes has only one non-vanishing element, its size-KN peak. The N ×K matrix whose columns are the codes of a binary complementary set, arranged in order, is the code set’s Complementary Code Matrix, or CCM. One might ask...

This chapter considers the structure of groups of operators preserving the aperiodic autocorrelation peak sidelobe level of mth-root codes. These groups are shown to be helpful for efficient enumeration of codes by peak sidelobe level for a given m and given code length N. In the binary case, it is shown that there is a single Abelian group of orde...

Results are presented for an adiabatic quantum algorithm to compute low peak sidelobe binary and unimodular codes on a D-Wave 2 quantum computer. The quantum algorithm is benchmarked against a conventional genetic algorithm (GA). The quantum algorithm shows roughly a 100 times speedup relative to the GA for binary codes longer that 100 bits and is...

New tools are developed for testing Complementary Code Matrix (CCM) existence, first for binary CCMs and then for more general p-phase classes, by formulating a CCM in terms of its row-correlation function. It is shown that a result of Lam and Leung on vanishing sums of roots of unity is useful in this approach for developing p-phase CCM existence...

The authors generalise the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having more than two codes, which they call Doppler-null codes. This is accomplished by exploiting number-theoretic results involving the sum-of-digits function and a generalisation to more than tw...

A set of unimodular (or binary) code vectors is complementary if the sum of the aperiodic autocorrelation sidelobes is zero, for every sidelobe. Complementary code sets are constructed using a matrix formulation in which code vectors form the columns of a matrix, called a Complementary Code Matrix, or CCM. Known construction methods for Hadamard ma...

A set of unimodular code vectors is complementary if the sum, over all code vectors, of the aperiodic autocorrelation sidelobes is zero. Complementary code sets are constructed using a matrix formulation in which code vectors form the columns of a matrix, called a complementary code matrix (CCM). Known construction methods for Hadamard matrices are...

Binary complementary code sets offer a possibility that single binary codes cannot-zero aperiodic autocorrelation sidelobe levels. These code sets can be viewed as the columns of so-called complementary code matrices, or CCMs. This matrix formulation is particularly useful in gaining the insight needed for developing an efficient exhaustive search...

This chapter considers the structure of groups of operators preserving the aperiodic autocorrelation peak sidelobe level of mth-root codes. These groups are shown to be helpful for efficient enumeration of codes by peak sidelobe level for a given m and given code length N. In the binary case, it is shown that there is a single Abelian group of orde...

Now is a good time to work on the boundaries of practice and theory, of art and science. We are seeing a rising tide of interest in these boundaries. Witness the growing Bridges movement, which has been exploring the connections between mathematics and the arts. Similarly, JoAnne Growney's blog, Intersections -- Poetry with Mathematics, explores th...

High resolution radar using linear frequency modulation (LFM) and non-linear frequency modulation (NLFM) transmit a wideband pulse to achieve a narrow range resolution. We consider digital multi-rate implementation of LFM and NLFM pulse compression. For the LFM case, we consider short-length fixed-point fast Fourier transform (FFT) within the multi...

Nearly optimal or globally optimal integrated sidelobe level (ISL) polyphase codes are found for lengths 46 through 80 by using a stochastic optimization technique. Polyphase Barker codes are found for lengths 64 to 70, and 72, 76, and 77 using constrained optimization starting at optimal-ISL codes.

Best-known binary code autocorrelation peak sidelobe levels (PSLs) are updated for lengths 71 to 105. For lengths 71 to 82, codes with PSL 4 are found, establishing 4 as almost certainly the optimal value for these lengths. PSL-5 codes are produced for all lengths from 83 to 105, in many cases improving on best-known values.

Low aperiodic-autocorrelation peak sidelobe levels (PSLs) relate to enhanced range resolution for binary-phase-coded radar and communication waveforms. Typical methods to identify the minimum-attainable PSL for a given code length N require exhaustive calculations whose computational burden grows exponentially with N. In this project, exact PSL his...

The work presented here describes an exhaustive search for minimum peak sidelobe level (PSL) binary codes, combining several devices for efficiency. These include combinatoric tree search techniques, the use of PSL-preserving symmetries to reduce search space, data representations and operations for fast sidelobe computation, and a partitioning sch...

The relationship between binary code imbalance and each of two measures of average autocorrelation sidelobe level is examined. For the first one, the arithmetic mean of signed sidelobes, an especially simple relationship is derived. It is found that the size of this average is minimized when imbalance is as dose as possible to the square root of th...

Cohen et al., (1990), found the lowest-PSL (peak sidelobe level) codes for N⩽48 using a search algorithm that exploits two PSL-preserving operations. This note explains how to exploit a third PSL-preserving operation in combination with the first two in a single exhaustive search routine. There is a two-fold payoff: a factor-of-two search speed...

The imbalance of a binary code with elements either +1 or -1 is the number of -1 elements subtracted from the number of +1 elements. A balanced code is one with 0 imbalance. Balanced codes are often used for pulse compression because the DC response of a linear pulse compression module is proportional to the imbalance. A nonzero DC bias combined wi...

For a given interval matrix, it would be valuable to have a practical method for determining the family of matrices which are inverses of its members. Since the exact family of inverse matrices can be difficult to find or to describe, effort is often applied to developing methods for determining matrix families with interval structure which "best"...

An interval matrix can be represented in terms of a center matrix and a nonnegative error matrix, specifying maximum elementwise perturbations from the center matrix. A commonly proposed robust stability (regularity) characterization for an interval matrix with a stable (nonsingular) center matrix identifies the minimum scaling of this error matrix...

Recently Rohn and Poljak proved that for interval matrices with rank-one radius matrices testing singularity is NP-complete. This paper will show that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the interval matrices, testing singularity reduces...

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