Designs and full-rank STBCs from DFT domain description of cyclic codes

Abstract

-Viewing an n length vector over Fqm as an m × n matrix over Fq, by expanding each entry of the vector with respect to a basis of Fqm over Fq, the rank weight of the n length vector over Fqm is the rank of the corresponding m×n matrix over Fq. It is known that under some conditions, n-length cyclic codes over Fqm , (n|q m −1 and m ≤ n) have full rank. In this paper, using this result we obtain a design using which we construct full-rank Space-Time Block Codes (STBCs) for m transmit antennas over signal sets matched to Fq where q = 2 or q is a prime of the form 4k + 1. We also propose a construction of STBCs using n-length cyclic codes over Fqm , for r transmit antennas, where r ≤ n and r|m. I. Extended SummaryThe characterization of cyclic codes using the Rank metric has been studied in [1]. In this paper, we use the main result of [1] to obtain full-rank STBCs.Definition 1 A rate-k/n, n × l linear design over a field F is an n × l matrix with all its entries F -linear combinations of k variables which are allowed to take values from the field F . Let (n, q) = 1 and n|q m − 1, where q is either 2 or a prime of the form 4k + 1. Let [j]n be a q-cyclotomic coset of In of size m. By restricting Aj, j ∈ [j]n, to Fqm and constraining all other transform components to zero, we have a n-length code over Fqm whose codewords are of the form,ˆA j α −j Aj α −2j Aj · · · α −(n−1)j Aj˜where α is a primitive n-th root of unity in Fqm and Aj ∈ Fqm . Viewing Aj as a m-length column vector over Fq, the codewords can be viewed as m × n matrices over Fq given by 2 6 6 6 4whereand β is a primitive element of Fqm . Notice that (1) is a design over Fq and the variables are allowed to take values from a signal set matched to Fq. Also, note that this is possible for any linear code, however only for cyclic codes we have information about the rank. To obtain an STBC from the above design, we have to map the elements of Fq into the complex field such that the rank is preserved. The following maps have been proposed for the same: Thus, by using the above map, we obtain STBCs over BPSK signal sets. Case 2. q = 4k + 1, Lusina et al. [3]: Let q be a prime of the form q = 4k + 1. Then, it is known that q = u 2 + v 2 for some integers u and v. Let Π = u − iv. Then w modulo Π of any integer w is defined as, ζ = w modulo Π = w − h wΠ ΠΠ i Π where [.] performs the operation of rounding to the nearest Gaussian number. In [3], it is proved that the Gaussian numbers modulo Π form a field, GΠ = {ζ0 = 0, ζ1 = 1, ζ2, · · · , ζq−1} where ζi = i mod Π. Example 1 Let the number of transmit antennas be 2 and q = 5. Then, we take n = 3 and m = 2. The 5-cyclotomic coset of 1 is {1, 2}. Thus, allowing only A1 to take values from F25 and constraining other components to zero, we have a full-rank 2 × 3 STBC with codewords of the formwhere a0, a1 ∈ F5 and ξ : F5 → G1+2i. STBCs for r transmit antennas, r|m: Let q be a prime of the form 4k + 1. Then, we have the map ξ : Fq → Gπ., where β is a root of a polynomial f irreducible over Fq and of degree r. The...