Kalle Ranto

On the Densest MIMO Lattices From Cyclic Division Algebras

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.

On the algebraic structure of the Silver code: A 2 × 2 perfect space-time block code

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.

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.

Asymmetric Space-Time Block Codes for MIMO Systems

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.

Maximal orders in space-time coding: Construction and decoding

Previously, it was 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 non-vanishing minimum determinant. In this paper, the actual procedure of constructing maximal orders is described in more detail, aiming to provide a handy tool also for researchers with only a modest mathematical background. For instance, it is explicitly shown, step by step, how to construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the famous Golden code of the same minimum determinant. In order to decode maximal order based space-time codes, the usual sphere decoder has to be modified. A pseudo algorithm describing the additional steps is given. For the algorithm to function it is essential that we also speed up the search for the shortest lattice vectors ensuring in this way that the usage of a codebook becomes feasible. Both the search and the decoding can be performed by adding an upper bound on the energy of the single vector in use.

On algebraic decoding of the Z/sub 4/-linear Goethals-like codes

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.

On Four-Valued Niho-Type Cross-Correlation Functions of

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

m

The quaternary Goethals-like codes are a family of Z4-linear codes of length 2m, m odd, which have 222m-3m-2 codewords and minimum Lee distance 8. They are further parameterized by an integer k subject to the constraint (k, m) = 1. In this note we give a simple new proof of the fact that the minimum distance of these codes does not depend on the auxiliary parameter k.

-Sequences

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.

A simple proof to the minimum distance of Z 4 -linear Goethals-like codes

We will completely describe the solutions of the equation (x+1)d=xd+1 in the field GF(q2), where q=pkand d is of Niho type, i.e., d≡1 (modq−1). Our results have applications in the theory of cross-correlation functions of m-sequences and in the theory of cyclic codes.