Catálogo de publicaciones - libros

We explore a connection between permutation polynomials of the form () and cyclotomic mapping permutation polynomials over finite fields. As an application, we characterize a class of permutation binomials in terms of generalized Lucas sequences.

Single-track Gray codes are cyclic Gray codes with codewords of length , such that all the tracks which correspond to the distinct coordinates of the codewords are cyclic shifts of the first track. These codes have advantages over conventional Gray codes in certain quantization and coding applications. We survey the main results on this problem, their connections to sequence design, and discuss some of the interesting open problems.

We study the asymptotic behavior of stream cipher security measures associated with algebraic feedback shift registers and feedback based on the ring . For non-periodic sequences we consider normalized -adic complexity and study the set of accumulation points for a fixed sequence. The the set of accumulation points is a closed subinterval of the real closed interval [0,1]. We see that this interval is of the form [,1 − ] “most” of the time, and that all such intervals occur for some sequence.

Two wideband physical-layer multiple access techniques for mobile telephony and data are compared: CDMA, which in one of several manifestations has been chosen for virtually all third generation cellular systems, and OFDM with MIMO, which seems to be the most favored for a future generation. We compare the Shannon capacities of the two and conclude that with appropriate processing CDMA and OFDM capacities are similar but that the latter may more effectively be combined with MIMO antennas to provide higher capacities.

One of the most important research areas in coding theory is . This is a large subject, but the basic problem is easily stated: determine or estimate the (, ..., ) for an (,) binary linear code, specified by a ( − ) × parity-check matrix with entries from (2). Here

where is the weight of the vector . If the number of codewords is large, the logarithmic weight enumerator, i.e.,

is more convenient. If a code belongs to a family of codes which share similar properties, the log-weight enumerator may approach a limiting function called the :

In modern coding theory, is usually very large and very sparse, e.g., the row and column sums are bounded as → ∞. The corresponding codes are called Often we are faced with large collections, or , of long LDPC codes, which share similar properties, in which case it may be difficult to find the spectral shape of an individual member of the ensemble, but relatively easy to calculate the .

In this short note we introduce two new sequences defined using the sum of digits in the representation of an integer in a certain base. A connection to Niven numbers is proposed and some results are proven.

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register (LFSR) of length and a Boolean filtering function of degree that combines bits from the shift register and creates an output bit at any time . A new attack on stream ciphers based on linear shift registers has recently been described by the authors in [3]. This attack is modified to stream ciphers based on any linear shift register and not only for LFSRs. The focal point of this paper is to present a linear description of the filter generator in terms of matrices. The filter generator is viewed entirely in terms of powers of a unique linear operator together with a vector representing the filtering function. It is proved that embodies the coefficient sequences described in [3]. Thus, interesting properties of the vector space (e.g. the dimension of the equation systems) generated by the filter generator can be analysed using theory of cyclic vector spaces, which very elegantly complements analysis in terms of the roots of the LFSR.

In this paper, we investigate the existence of modular sonar sequences of length and mod where is a product of twin primes. For  = 3·5 = 15, we have found some old and new examples by exhaustive search. However, the very next case  = 5·7 = 35 is completely open, in that neither we know (have) an example, nor we prove the nonexistence. We describe simply some approach to locate a single example of modular sonar sequences of length 35 and mod 35, assuming (or hoping) that one exists.

Any span sequences can be regarded as filtering sequences. From this observation, new randomness criteria for span sequences are proposed. It is proved that the feedback function of a span sequence can be represented as a composition of its trace representation, or equivalently, its discrete Fourier transform, and a permutation from the state space of the sequence to the multiplicative group of the finite field (2), and vice versa. Significant enhancements for randomness of span sequences, so that de Bruijn sequences, are illustrated by some examples.

The filter generator consists of a linear feedback shift register (LFSR) and a Boolean filtering function that combines some bits from the shift register to create a key stream. A new attack on the filter generator has recently been described by Rønjom and Helleseth [6]. This paper gives an alternative and extended attack to reconstruct the initial state of the LFSR using the underlying subspace structure of the filter sequence. This improved attack provides further insight and more flexibility in performing the attack by Rønjom and Helleseth. The main improvements are that this attack does not use the coefficient sequences that were fundamental in the previous attack and also works in the unlikely cases when the original attack needed some modifications.