encoding, power,

Abstract not found

The characterization of perfect single-error correcting codes, or 1-perfect codes, is an open question for a long time. Recently, J. Rifà has proved that a binary 1-perfect code can be viewed as a distance compatible structure in F n and an homomorphism ` : F n \Gamma! \Omega\Gamma where\Omega is a loop (a quasigroup with identity element). In this paper, we study 1-perfect codes in the extremal case when F n , with the distance compatible structure, and\Omega are Abelian groups. More precisely, we study 1-perfect codes which are subgroups of F n with a distance compatible Abelian structure. We compute the set of admissible parameters for such codes, and we give a construction for any case. We prove that two such codes are different if they have different parameters. The resulting codes are always systematic, and we prove their unicity. Therefore, we are giving a full characterization. Easy coding and decoding algorithms are also presented.

In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of error-correcting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraic-geometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert–Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory.

We present an efficient algorithm for computing the minimal trellis for a group code over a finite Abelian group, given a generator matrix for the code. We also show how to cornpure a succinct representation of the minimal trellis for such a code, andpresent algorithms that use this information to efficiently compute local descriptions of the minimal trellis. This extends the work of Kschischang and Sorokine, who handled the case of linear codes over fields. An important application of our algorithms is to the construction qf minireal trellises for lattices. A key step in our work is handling codes over cyclic groups C'p, where p is a prime. Such a code can be viewed as a submodule over the ring Zp. Because of the presence of zero-divisors in the ring, submodules do not share the useful properties of vector spaces. We get around this difficulty by restricting the notion of linear combination to p-linear combination, and introducing the notion of a p-generator equence, which enjoys properties similar to that of a generector matrix for a vector space.

Abstract not found

There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z4–linear Kerdock and Preparata codes on the other. These inter– relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non–isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z4–linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of “boring ” affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes “boring ” in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.

Space–time coding has been studied extensively as a powerful error correction coding for systems with multiple transmit antennas. An important design goal is to maximize the level of space diversity that a code can achieve. Toward this goal, the only systematic algebraic coding theory so far is binary rank theory by Hammons and El Gamal for binary phase-shift keying (BPSK) modulated codes defined over binary field and quaternary phase-shift keying (QPSK) modulated codes defined over modulo four finite ring. To design codes with higher bandwidth efficiency, we develop an algebraic rank theory to ensure full space diversity for PP quadrature and amplitude modulated (QAM) codes for any positive integer. The theory provides the most general sufficient condition of full space diversity so far. It includes the BPSK binary rank theory as a special case. Since the condition is over the same domain that a code is defined, the full space diversity code design is greatly simplified. The usefulness of the theory is illustrated in examples, such as analyses of existing codes, constructions of new space–time codes with better performance, including the full diversity space–time turbo codes.

codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over Z4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over Z4, but Hamming codes in general are not, nor is the Golay code. 1.

A code C is Z2Z4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper Z2Z4-additive codes are studied. Their corresponding binary images, via the Gray map, are Z2Z4-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for Z2Z4-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of Z2Z4-additive codes are given.