It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory, a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving the bound are given. For example, a matrix lattice with quadrature amplitude modulation (QAM) coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant is constructed. Also, a general algorithm due to Ivanyos and Ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed. Also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed.

Recently, a family of full-rate, full-diversity space-time block codes (STBCs) for 2 times 2 multiple-input multiple-output (MIMO) channels was proposed in the works of Tirkkonen et al., using a combination of Clifford algebra and Alamouti structures, namely twisted space-time transmit diversity code. This family was recently rediscovered by Paredes et al., and they pointed out that such STBCs enable reduced-complexity maximum-likelihood (ML) decoding. Independently, the same STBCs were found in the work of Samuel and Fitz (2007) and named multi-strata space-time codes. In this paper we show how this code can be constructed algebraically from a particular cyclic division algebra (CDA). This formulation enables to prove that the code has the non-vanishing determinant (NVD) property and hence achieves the diversity-multiplexing tradeoff (DMT) optimality. The fact that the normalized minimum determinant is 1/radic(7) places this code in the second position with respect to the golden code, which exhibits a minimum determinant of 1/radic(5), and motivates the name silver code.

In this paper, the need for the construction of multiple input-double output (MIDO) space-time block codes (STBCs) is discussed, concentrating on the case of four transmitters for simplicity. Above the trivial method, i.e. switching off the extra layers in the usual multiple input-multiple output (MIMO) setting, two smarter yet simple MIDO construction methods are proposed. The use of these general methods is then demonstrated by building explicit, sphere decodable codes using two different cyclic division algebras (CDAs). We verify by computer simulations that the newly proposed methods perform extremely well as opposed to the trivial construction.

In this paper, the need for the construction of asymmetric space-time block codes (ASTBCs) is discussed, mostly concentrating on the case of four transmitting and two receiving antennas for simplicity. Above the trivial puncturing method, i.e. switching off the extra layers in the symmetric multiple input-multiple output (MIMO) setting, a more sophisticated yet simple asymmetric construction method is proposed. This method can be converted to produce multi-block space-time codes that achieve the diversity-multiplexing (D-M) tradeoff. It is also shown that maximizing the density of the newly proposed codes is equivalent to minimizing the discriminant of a certain order. The use of the general method is then demonstrated by building explicit, sphere decodable codes using different cyclic division algebras (CDAs). We verify by computer simulations that the newly proposed method can compete with the puncturing method, and in some cases outperforms it. Our conquering construction exploiting maximal orders improves even upon the punctured perfect code and the DjABBA code.

We show why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed non-vanishing minimal determinant. Using results from class field theory we derive a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas). We also give examples of division algebras achieving our bound. For example, we construct a matrix lattice with QAM coefficients that has (inside 'large' subsets of the signal space) 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We also give another matrix lattice with coefficients from the hexagonal lattice with an even higher density

Consider the cross-correlation function Cd(tau) between two m-sequences of period p2k-1 that differ by a decimation d. Assume that d is of Niho type and Cd(tau) is four-valued. In this correspondence, the values of Cd(tau) are described. If p=2 then the values of Cd(tau) are -1-2k,-1,-1+2k, and -1+2k+jfor some integer j>0 for which 2j divides k. If p>2 then the values are -1-pk,-1,-1+pk, and -1+2middotpk

We construct several new families of simple 3-designs from codewords of the $\mathbf Z_4$-Goethals codes. These designs have parameters 3-$(2^m,8,\lambda)$ with odd $m\ge 5$. The smallest design has $\lambda=14(2^m-8)/3$, and the others are corollaries of this and some previously known designs. In the existence proofs we analyze the low-weight codewords and count the number of solutions to certain systems of equations over finite fields.

The Z/sub 4/-linear Goethals-like code of length 2/sup m/ has 2/sup 2m+1-3m-2/ codewords and minimum Lee distance 8 for any odd integer m/spl ges/3. We present an algebraic decoding algorithm for all Z/sub 4/-linear Goethals-like codes C/sub k/ introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations.