A general approach to construction and determination of the linear complexity of sequences based on cosets

We give a general approach to $N$-periodic sequences over a finite field \F_q constructed via a subgroup $D$ of the group of invertible elements modulo $N$. Well known examples are Legendre sequences or the two-prime generator. For some generalizations of sequences considered in the literature and for some new examples of sequence constructions we determine the linear complexity

Uniform determinantal representations

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.Comment: 23 pages, 3 figures, 4 table

Design and Analysis of a Trellis-Based Syndrome Distribution Matching Algorithm

Negli ultimi anni, la tecnica di modulazione chiamata "probabilistic shaping" ha assunto grande interesse sia in ambito accademico che industriale. Quando i bit vengono modulati e trasmessi come simboli di canale di comunicazione, è necessario tenere in considerazione che distribuzioni di ingresso uniformi non consentono in molti casi di avvicinarsi alla capacità. La tecnica di probabilistic shaping consente quindi di sfruttare in modo più efficiente il canale di trasmissione. L'elemento fondamentale che consente di implementare il probabilistic shaping è noto come "distribution matcher". Nel capitolo 1 viene presentata una panoramica generale sul modello di un sistema di comunicazione digitale. Nel capitolo 2 viene discusso il distribution matching e viene introdotta la tecnica constant composition distribution matcher (CCDM). Nel capitolo 3 si descrive l’algoritmo di distribution matching proposto nella tesi. Nei capitoli 4 e 5 si presentano i risultati di due scenari differenti, proponendo anche possibili applicazioni in cui risulta utile l’algoritmo. In conclusione, si forniscono osservazioni e possibili miglioramenti

Error control techniques for satellite and space communications

Shannon's capacity bound shows that coding can achieve large reductions in the required signal to noise ratio per information bit (E sub b/N sub 0 where E sub b is the energy per bit and (N sub 0)/2 is the double sided noise density) in comparison to uncoded schemes. For bandwidth efficiencies of 2 bit/sym or greater, these improvements were obtained through the use of Trellis Coded Modulation and Block Coded Modulation. A method of obtaining these high efficiencies using multidimensional Multiple Phase Shift Keying (MPSK) and Quadrature Amplitude Modulation (QAM) signal sets with trellis coding is described. These schemes have advantages in decoding speed, phase transparency, and coding gain in comparison to other trellis coding schemes. Finally, a general parity check equation for rotationally invariant trellis codes is introduced from which non-linear codes for two dimensional MPSK and QAM signal sets are found. These codes are fully transparent to all rotations of the signal set

Coherence Optimization and Best Complex Antipodal Spherical Codes

Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases