1 A Simple Construction of Nonvanishing Determinant Space-Time Block Codes Based on Cyclic Division Algebra Xiaoyong Guo and Xiang-Gen Xia Abstract— Cyclic division algebra (CDA) has recently become a major technique to construct nonvanishing determinant (NVD) space-time block codes. The CDA based construction method usually consists of two steps. The first step is to construct a degree-n cyclic extension over a base field and the second step is to find a non-norm algebraic integer in the base field. In this paper, we first propose a simple construction method for cyclic extensions and then propose an elementary condition for non- norm elements for QAM and HEX signal constellations. Design examples are shown for n =2 to n = 20, where n is the number of transmit antennas, and it is shown that with our newly proposed construction, non-norm elements with smaller absolute values than the existing ones can be found. I. I NTRODUCTION Space-time block codes (STBC) with nonvanishing deter- minant (NVD) have attracted much attention lately, see for example [1]–[14]. In particular, Elia et. al. [6] have shown that full rate STBC with NVD achieve the diversity-multiplexing tradeoff obtained by Zheng-Tse [15]. There are two major methods to construct full rate STBC with NVD. One is to use the multi-layer (threaded) structure, see for example [4], [9], [10], [13], [14] and the other is to use the cyclic division algebra (CDA) structure that was first used to construct full diversity STBC in [16], see for example [5]–[8], [11], [12]. This paper is only interested in the CDA approach. In [5], Kiran and Rajan presented a general construction of CDA-based NVD STBC for a class of n: n =2 m , 2 · 3 m , 3 · 2 m , and n = q k (q − 1), q is prime and q = 3 mod 4, where n is the number of transmit antennas. In [6], Elia et. al. presented a more general construction of NVD STBC based on CDA for any n and all of them are for QAM signals (similar constructions were also obtained for HEX signals in Z[j]). In this paper, we propose a simple construction method, which is easy to implement on a computer. The CDA construction usually consists of two steps. The first step is to construct a degree-n cyclic extension over a base field and the second step is to find a non-norm algebraic integer in the base field. For the first step of the construction, we propose a simple construction method by using the Kronecker-Weber Theorem that implies that any cyclic extension K over Q is a subfield of some cyclotomic field. Then, i or j is properly added to The authors are with the Department of Electrical and Computer En- gineering, University of Delaware, Newark, DE 19716. Email: {guo, xxia}@ee.udel.edu. Their work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-05-1-0161, the National Science Foundation under Grant CCR-0325180. K to make it a cyclic extension over Q(i) or Q(j). For the second step, based on Kiran and Rajan’s sufficient condition for a non-norm element, we develope an elementary condition for non-norm elements that is easy to check. This paper is organized as follows. In Section 2, we give a brief description of the CDA-based NVD STBC construction. In Section 3, we present a construction of cyclic extensions. In Section 4, we present an elementary condition for non- norm elements. In Section 5, we present some design examples and some comparison with the existing codes. Throughout this paper, we use Z and Q to denote integer ring and rational field, respectively, i = √ −1 and j = exp( i2π 3 ), and ξ can be either i or j. II. STBC BASED ON CYCLIC DIVISION ALGEBRA A cyclic algebra A over a number field F is determined by 1) a degree-n cyclic extension L/F, i.e., Galois group Gal(L/F)= 〈σ〉 is cyclic; 2) a γ ∈ F *  F\{0}. Every element in A can be represented by a matrix in the following form, C =          x 0 γσ(x n-1 ) ··· γσ n-1 (x 1 ) x 1 σ(x 0 ) ··· γσ n-1 (x 2 ) x 2 σ(x 1 ) ··· γσ n-1 (x 3 ) . . . . . . . . . . . . x n-2 σ(x n-3 ) ··· γσ n-1 (x n-1 ) x n-1 σ(x n-2 ) ··· σ n-1 (x 0 )          , (1) where x l ∈ L,l =0, 1,...,n − 1. If γ l ∈ N L/F (L), i.e., γ l =  n-1 j=0 σ j (x) for any x ∈ L, l =1, 2,...,n − 1, then the cyclic algebra A is a division algebra, i.e., every non-zero element in A has a multiplicative inverse. The above condition imposed on γ is called norm condition.A γ satisfying norm condition is said to be a non-norm element [6], [16], [17]. We always have det(C) ∈ F, a concise proof is given in [6]. And we also have that det(C)=0 if and only if x l =0 for all l, i.e., code {C} has full diversity. If we choose F = Q(ξ) and x l ,l =0, 1,...,n − 1, to be algebraic integers in L with  n-1 l=0 x l =0, in addition, we choose a γ ∈ Z[ξ] which satisfies the norm condition, then det(C) is clearly a nonzero algebraic integer in Q(ξ), i.e., det(C) ∈ Z[ξ]\{0}. Therefore, we have | det(C)|≥ 1. This division algebra property gives us a way to construct NVD STBC [5]. Let e l ∈O L ,l =