José Carmelo Interlando

A classification of all totally real subfields K of cyclotomic field Q(zeta_{2^r}), for any r ≥ 4, and the fully-diverse related versions of the Z^n-lattice are presented along with closed-form expressions for their minimum product distance. Any totally real subfield K of Q(zeta_{2^r}) must be of the form K=Q(zeta_{2^2} + zeta_{2^2}^{-1}), where s = r − j for some 0 ≤ j ≤ r − 3. Signal constellations for transmitting information over both Gaussian and Rayleigh fading channels (which can be useful for mobile communications) can be carved out of those lattices.

Ongoing research and experiments have enabled quantum memory to realize the storage of qubits. On the other hand, interleaving techniques are used to deal with burst of errors. Effective interleaving techniques for combating burst of errors by using classical error-correcting codes have been proposed in several articles found in the literature, however, to the best of our knowledge, little is known regarding interleaving techniques for combating clusters of errors in topological quantum error-correcting codes. Motivated by that, in this work, we present new three and four-dimensional toric quantum codes which are featured by lattice codes and apply a quantum interleaving method to such new three and four-dimensional toric quantum codes. By applying such a method to these new codes we provide new three and four-dimensional quantum burst-error-correcting codes. As a consequence, new three and four-dimensional toric and burst-error-correcting quantum codes are obtained which have better information rates than those three and four-dimensional toric quantum codes from the literature. In addition to these proposed three and four-dimensional quantum burst-error-correcting codes improve such information rates, they can be used for burst-error-correction in errors which are located, quantum data stored and quantum channels with memory.

Let p be prime, m a positive integer( (m >= 1, and m >= 2 if p = 2), and chi(n) a multiplicative complex character on F-pm* with order n vertical bar (p(m)-1). We show that a partition {A(1),A(2),..., A(n)} of F-pm*( )is the partition by fibers of chi(n) if and only if these fibers satisfy certain additive properties. This is equivalent to showing that the set of multivariate characteristic polynomials of these fibers, completed with the constant polynomial 1, is the basis of an (n +1) -dimensional commutative algebra with identity in the ring Q[x(1),..., x(n)]/< x(1)(p)-1,...,x(n)(p)-1 >.

In this paper we generalize the ball-collision algorithm by Bernstein, Lange, Peters from the binary field to a general finite field. We also provide a complexity analysis and compare the asymptotic complexity to other generalized information set decoding algorithms.

A method for constructing rotated Z^n-lattices, with n a power of 2, based on totally real subfields of the cyclotomic field Q(\zeta_{2^r}), where r\geq 4 is an integer, is presented. Lattices exhibiting full diversity in some dimensions n not previously addressed are obtained.