Reginaldo Palazzo

Graphs, tessellations, and perfect codes on flat tori

Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney . Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered.

Constructions of new families of nonbinary CSS codes

New families of good q-ary (q is an odd prime power) Calderbank-Shor-Steane (CSS) quantum codes derived from two distinct classical Bose-Chaudhuri-Hocquenghem (BCH) codes, not necessarily self-orthogonal, are constructed. These new families consist of CSS codes whose parameters are better than the ones available in the literature and comparable to the parameters of quantum BCH codes generated by applying the q-ary Steanes enlargement of CSS codes.

A concatenated [(4, 1, 3)] quantum convolutional code

We construct a concatenated [(4, 1, 3)] quantum convolutional code from a (2, 1, 2) classical convolutional code. This quantum convolutional code can correct one general error.

Lattice constellations and codes from quadratic number fields

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric module a two-dimensional (2-D) grid, in particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

Construction and decoding of BCH codes over finite commutative rings

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t ,l etA0 ⊂ A1 ⊂ · ·· ⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂ · ·· ⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A ∗ and K ∗ are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A ∗ and K ∗ for each i ,w here 0≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Using combinatorial optimization to design good unit-memory convolutional codes

A method for designing good unit-memory convolutional codes is presented. The method is based on the decomposition of the original problem into two easier subproblems that can be formulated as optimization problems and solved by efficient heuristic search algorithms. The efficacy of this method is demonstrated by a table containing 33 new unit-memory convolutional codes (n,k) with 5 >

Signal constellations in the hyperbolic plane: A proposal for new communication systems

Signal constellations in the hyperbolic plane are provided as an alternative to traditional signal constellations in the Euclidean plane, since channels may actually exist for which the latter signal constellations are not as suitable as the former. A hyperbolic gaussian probability density function, based solely on geometrical considerations, is derived to determine the performance of the hyperbolic signal constellations. Benefits result from an approach conceived in terms of reduced signal-to-noise ratio, needed to achieve a prescribed error rate and equivalent optimum receiver complexity.

A network flow approach to convolutional codes

Presents an alternative way of looking at convolutional codes as a network of flows. From this association, previous known classes of nonsystematic convolutional codes are reproduced. The authors also present a decomposition of the problem of finding good convolutional codes into well known and structured problems such as the maximum flow problem, the knapsack problem, and the modular representation of block codes. A spanning tree search algorithm is also proposed for finding good codes. The authors consider the class of nonsystematic convolutional codes for which the encoders satisfy an equivalent property related to the feasibility of flow. Finally, extensive tables show the new and the previous known codes. >

A time-varying convolutional encoder better than the best time-invariant encoder

A periodically-time-varying binary (3, 2) convolutional code with period T=2 and memory M=1 having larger free distance than the best time-invariant (3, 2) convolutional code with M=1 is presented. >

Topological quantum codes on compact surfaces with genus g≥2

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.