In this paper, we address the problem of the universal coding of natural numbers. A new numeration system is introduced, which is based on variable-r meta-Fibonacci sequences and it is a generalization of the Zeckendorf numeration system. This new numeration system is used to construct binary, prefix-free, uniquely decodable universal codes called meta-Fibonacci codes. The main advantage of these codes is that they are parametrized by a sequence of numbers, the sequence o. By controlling the growth of the values of this sequence, we can control the length of the code word. This means that we can provide a general framework for building efficient universal coders for natural numbers. Such framework is applied to the upper bounds of the code word length defined by Leung-Yan-Cheong and Cover (1978), Levenshtein (1968), and Ahlswede (1997). There is no other code meeting these bounds. In each case, we build meta-Fibonacci codes and demonstrate that the upper bound of their code word length is satisfied up to an additive constant, thereby solving these open problems. The framework may be applied to other upper bounds that satisfy Kraft inequality.

In this paper, we introduce fractional number-theoretic transforms (FrNTT) based on matrix functions. The approach, which is a kind of finite field extension of the method presented in [1], does not require the construction of any number-theoretic transform eigenvector set. In this sense, the definition presented in this work is simpler than that of another recently introduced FrNTT. An image encryption scheme based on the proposed FrNTT is suggested.

In this paper, a multiuser communication technique based on the discrete fractional Fourier transform (DFrFT) is discussed. Eigenvectors of the DFrFT transform matrix are used as user sequences, which are transmitted over a real adder channel. Compared to other transforms used in the same context, the advantage of using the DFrFT is that the sequences can be generated from a systematic procedure and the number of generated subspaces, which determines the maximum number of simultaneous users of a scheme, is arbitrary. After describing the basic idea of our approach, we discuss some aspects related to its practical implementation and present preliminary simulation results.

In this paper, a finite field version for the fractional Fourier transform is introduced. We show that, in some aspects, the finite field fractional Fourier transform (4FT) is in perfect analogy with the discrete fractional Fourier transform. On the other hand, we consider some definitions and properties from the finite field context which allow us to discuss particularities of the 4FT.

In this paper, a new definition of Chebyshev polynomials over prime finite fields is introduced. Our approach uses a finite field trigonometry and reveals some aspects concerning the security of a recently proposed public-key encryption algorithm based on those polynomials. Particularly, we show that recovering the corresponding plaintext from a given ciphertext involves the discrete logarithm problem.