Infinite families of cyclic and negacyclic codes supporting 3-designs

Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over \gf(q) supporting an infinite family of $3$-designs and two infinite families of negacyclic codes over \gf(q^2) supporting two infinite families of $3$-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over \gf(q) are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF($4$) supporting a $4$-design and six open problems are also presented in this paper

On the parameters of extended primitive cyclic codes and the related designs

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight of this code supporting 2-designs. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined. The parameters of the related $2$-designs are also given. Moreover, we prove that the codewords with minimum Hamming weight supporting 3-designs, which gives an affirmative solution to Heng's conjecture

Interference-Mitigating Waveform Design for Next-Generation Wireless Systems

A brief historical perspective of the evolution of waveform designs employed in consecutive generations of wireless communications systems is provided, highlighting the range of often conflicting demands on the various waveform characteristics. As the culmination of recent advances in the field the underlying benefits of various Multiple Input Multiple Output (MIMO) schemes are highlighted and exemplified. As an integral part of the appropriate waveform design, cognizance is given to the particular choice of the duplexing scheme used for supporting full-duplex communications and it is demonstrated that Time Division Duplexing (TDD) is substantially outperformed by Frequency Division Duplexing (FDD), unless the TDD scheme is combined with further sophisticated scheduling, MIMOs and/or adaptive modulation/coding. It is also argued that the specific choice of the Direct-Sequence (DS) spreading codes invoked in DS-CDMA predetermines the properties of the system. It is demonstrated that a specifically designed family of spreading codes exhibits a so-called interference-free window (IFW) and hence the resultant system is capable of outperforming its standardised counterpart employing classic Orthogonal Variable Spreading Factor (OVSF) codes under realistic dispersive channel conditions, provided that the interfering multi-user and multipath components arrive within this IFW. This condition may be ensured with the aid of quasisynchronous adaptive timing advance control. However, a limitation of the system is that the number of spreading codes exhibiting a certain IFW is limited, although this problem may be mitigated with the aid of novel code design principles, employing a combination of several spreading sequences in the time-frequency and spatial-domain. The paper is concluded by quantifying the achievable user load of a UTRA-like TDD Code Division Multiple Access (CDMA) system employing Loosely Synchronized (LS) spreading codes exhibiting an IFW in comparison to that of its counterpart using OVSF codes. Both system's performance is enhanced using beamforming MIMOs

Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming

Elfving's Theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of $c-$optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, $c-,A-,T-$ and $D-$optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, our SOCP approach can deal with design problems in which the variable is subject to several linear constraints. We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse $c-$optimal design always has a solution which is a matrix of rank $1$. Therefore, the complexity of this problem fades. We also investigate a \emph{model robust} generalization of $c-$optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments. When the number of unknown parameters is small, or when the number of linear functions of the parameters to be estimated is small, we show by numerical examples that our approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms