On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

AbstractWe construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason–MacWilliams group. We find this canonical set in the vector space (⊗i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C [ x, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

OBSERVATION ON THE WEIGHT ENUMERATORS FROM CLASSICAL INVARIANT THEORY

The purpose of this paper is to collect computations related to the weight enumerators and to present some relationships among invariant rings.The latter is done by combining two maps,the Brou\'e-Enguehard map and Igusa's \rho homomorphism. For the completeness of the story,some formulae are given which are not necessarily used in the present manuscript.Sections 1 and 2 contain no new result

Jacobi polynomials and design theory II

In this paper, we introduce some new polynomials associated to linear codes over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We also give the notion of generalized $q$-colored $t$-designs. As an application of the generalized $q$-colored $t$-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over $\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.Comment: 28 page

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Forward and Inverse Modeling of GPS Multipath for Snow Monitoring

Snowpacks provide reservoirs of freshwater, storing solid precipitation and delaying runoff to be released later in the spring and summer when it is most needed. The goal of this dissertation is to develop the technique of GPS multipath reflectometry (GPS-MR) for ground-based measurement of snow depth. The phenomenon of multipath in GPS constitutes the reception of reflected signals in conjunction with the direct signal from a satellite. As these coherent direct and reflected signals go in and out of phase, signal-to-noise ratio (SNR) exhibits peaks and troughs that can be related to land surface characteristics. In contrast to other GPS reflectometry modes, in GPS-MR the poorly separated composite signal is collected utilizing a single antenna and correlated against a single replica. SNR observations derived from the newer L2-frequency civilian GPS signal (L2C) are used, as recorded by commercial off-the-shelf receivers and geodetic-quality antennas in existing GPS sites. I developed a forward/inverse approach for modeling GPS multipath present in SNR observations. The model here is unique in that it capitalizes on known information about the antenna response and the physics of surface scattering to aid in retrieving the unknown snow conditions in the antenna surroundings. This physically-based forward model is utilized to simulate the surface and antenna coupling. The statistically-rigorous inverse model is considered in two parts. Part I (theory) explains how the snow characteristics are parameterized; the observation/parameter sensitivity; inversion errors; and parameter uncertainty, which serves to indicate the sensing footprint where the reflection originates. Part II (practice) applies the multipath model to SNR observations and validates the resulting GPS retrievals against independent in situ measurements during a 1-3 year period in three different environments - grasslands, alpine, and forested. The assessment yields a correlation of 0.98 and an RMS error of 6-8 cm, with the GPS under-estimating in situ snow depth by approximately 15%. GPS daily site averages were found effective in mitigating random noise without unduly smoothing the sharp transitions as captured in new snow events. This work corroborates the readiness of quality-controlled GPS-MR for snow depth monitoring, reinforcing its maturity for operational usage