Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code - a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values - and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.Comment: 25 pages, no figure

Spectral approach to linear programming bounds on codes

We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in the projective spaces.Comment: 11 pages, submitte

Lower bounds for measurable chromatic numbers

The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10,..., 24, and we give a new proof that it grows exponentially with the dimension.Comment: 18 pages, (v3) Section 8 revised and some corrections, to appear in Geometric and Functional Analysi

Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.Comment: 30 pages, 3 table

Location and general characteristics of sediment deposits into man-entry combined sewers

International audienceThis study has the following principle aims. Identifying the contextual elements of sediment deposits which appear as the most important, or, in a way, as required conditions for clogging. It emerges that a weak slope is not a sufficient condition. We show that the main vulnerable sections are: downstream abrupt changes of geometry (the upper ramifications of man-entry sewers, the slope breakdowns,...) downstream flow losses (diffluence. overflows) under downstream influences (confluence, pumping stations,...). Clearing the general characteristics of these sediment deposits in French man-entry combined sewers. The mean D50 varies from 1 mm (Paris) to more than 2 mm. The finest deposits are rather specifically located (upstream of local obstacles, or under downstream influences during dry weather - for instance near a pumping station, in very flat sewers - or downstream from overflows). Evaluating the representativeness of the results from more accurate studies carried out in particular places, for instance the Marseille No 13 trunk. Drawing inferences about knowledge improvements and technical means.This study has the following principle aims. - Identifying the contextual elements of sediment deposits which appear as the most important, or, in a way, as required conditions for clogging. It emerges that a weak slope is not a sufficient condition. We show that the main vulnerable sections are: downstream abrupt changes of geometry (the upper ramifications of man-entry sewers, the slope breakdowns, ...) downstream flow losses (diffluence, overflows) under downstream influences (confluence, pumping stations, ...). - Clearing the general characteristics of these sediment deposits in French man-entry combined sewers. The mean D50 varies from 1 mm (Paris) to more than 2mm. The finest deposits are rather specifically located (upstream of local obstacles, or under downstream influences during dry weather - for instance near a pumping station, in very flat sewers -, or downstream from overflows). - Evaluating the representativeness of the results from more accurate studies carried out in particular places, for instance the Marseille No 13 trunk. - Drawing inferences about knowledge improvements and technical means

Characterization of Suspended-Solids in Urban Wet Weather Discharges

International audienceThe objective of this study is to reach a better understanding of the main characteristics of urban stormwater discharge pollution by providing information on: particulate pollution, its importance in relation to dissolved pollution and its abatement through decantation, the grain size, the specific mass and the settling velocity of these solids. The first part of this paper is given over to the presentation of the sites chosen and of the experimental methods put into operation. In the second part we go on to analyze the results obtained for ten rainfall events over four different sites, according to their geographical position, to the type of network and to the size of the catchment basin. It is quite clear from the results obtained that the solids are the main vector of pollution. These solids are very fine (median diameter = 25 to 44 mum) and generally have a very high settling velocity (median settling velocity = 4 to 11 m/h). Thus it can be slated that decantation represents an efficient method to combat the pollution in urban stormwater discharge and this has been proved experimentally

Evaluation of urban runoff volume to be treated at the outlet of separately sewered catchments

International audienceThe purpose of this study was to estimate the volume of urban stormwater which must be treated in order to achieve a given efficiency in terms of pollutant retention in a storm basin. The data used for this were those from the French national programme (1980-82), collected in four separately sewered catchments whose area is less than 100 hectares. The intercepted loads for one year and for the most polluted events have been calculated for interception volumes varying between 0 and 300 m3 per impervious hectare. It was concluded that 100 m3/imp ha can intercept about 80% of the annual load of SS, BOD5 and COD. But for the most polluted events, to achieve the same efficiency (80% of the produced load), a reservoir volume of 200 m3/imp ha is needed. The reduction of frequency of the most polluted events was also studied: for the events of which the load is m excess of 5% of the annual load produced, even a storage volume of 200 m3/imp ha is insufficient to entirely prevent these occurring.The purpose of this study was to estimate the volume of urban stormwater which must be treated in order to achieve a given efficiency in terms of pollutant retention in a storm basin. The data used for this were those from the French national programme (1980-82), collected in four separately sewered catchments whose area is less than 100 hectares. The intercepted loads for one year and for the most polluted events have been calculated for interception volumes varying between 0 and 300 m3 per impervious hectare. It was concluded that 100 m3/imp ha can intercept about 80% of the annual load of SS, BOD5 and COD. But for the most polluted events, to achieve the same efficiency (80% of the produced load), a reservoir volume of 200 m3/imp ha is needed. The reduction of frequency of the most polluted events was also studied: for the events of which the load is in excess of 5% of the annual load produced, even a storage volume of 200 m3/imp ha is insufficient to entirely prevent these occurring

Des décanteurs pour traiter la pollution des rejets urbains par temps de pluie

Dans cette communication nous présentons, tout d'abord, les premiers résultats de nos travaux sur le dimensionnement des ouvrages de traitement par décantation des rejets urbains par temps de pluie. Il s'agit plus précisément : - des volumes et des débits à traiter pour être efficace dans la lutte contre la pollution des rejets urbains par temps de pluie ; - des seuils de vitesse de chute permettant d'évaluer la capacité théorique d'abattement de cette pollution. Nous étudions ensuite et très rapidement quelques éléments complémentaires permettant d'améliorer la conception des ouvrages de décantation