Maximum rank distance codes as space~time codes

Lusina, P.; Gabidulin, E.M.; Bossert, Martin

doi:10.1109/tit.2003.818023

Cited by 97 publications

(82 citation statements)

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“…We construct optimal STBCs for any given value of transmit diversity gain d ≤ n t ≤ l. Our construction specializes to the technique used in [10] for a = f = 1 and p = 4k + 1, and our results also prove that the map used in [10] is not only full-rank preserving (d = n t ) but also preserves any arbitrary rank (d < n t ). The contributions of this paper are:…”

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 87%

“…In this paper, we generalize the results in [10] by working in a more general setting; we construct STBCs from codes over Galois ring GR(p a , f ). A Galois ring GR(p a , f ) is a ring isomorphic to the residue class ring Z p a /(g(x)), where Z p a is the integer ring modulo p a and f (x) is a monic basic irreducible polynomial over Z p a , of degree f [11].…”

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 88%

“…In subsequent works [4,7,8,9], Lu and Kumar have generalized this construction to the so called Generalized Unified (GU) construction, with a map from F nt×l p to S nt×l , where p is any prime (including 2) and S is a subset of the ring Z[ω p K ]; ω p K is a primitive p K th root of unity. In a related work [10], full-rank MRD codes over F p where p is of the form 4k + 1 have been used for constructing space-time codes, which involves:…”

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 99%

“…• STBCs with optimal rate-diversity tradeoff are constructed using codes over arbitrary Galois rings, which generalizes the map in [10].…”

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 99%

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Optimal STBCs from codes over galois rings

Kiran

Rajan

2005 IEEE International Conference on Personal Wireless Communications, 2005. ICPWC 2005.

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A Space-Time Block Code (STBC) C ST is a finite collection of n t × l complex matrices. If S is a complex signal set, then C ST is said to be completely over S if all the entries of each of the codeword matrices are restricted to S. The transmit diversity gain of such a code is equal to the minimum of the ranks of the difference matrices (X − X ), for any X = X ∈ C ST , and the rate is R = log |S| |CST | l complex symbols per channel use, where |C ST | denotes the cardinality of C ST . For a STBC completely over S achieving transmit diversity gain equal to d, the rate is upper-bounded as R ≤ n t − d + 1. An STBC which achieves equality in this tradeoff is said to be optimal. A Rank-Distance (RD) code C F F is a linear code over a finite field F q , where each codeword is a n t ×l matrix over F q . RD codes have found applications as STBCs by using suitable rank-preserving maps from F p to S. In this paper, we generalize these rank-preserving maps, leading to generalized constructions of STBCs from codes over Galois ring GR(p a , k). To be precise, for any given value of d, we construct n t × l matrices over GR(p a , k) and use a rank-preserving map that yields optimal STBCs with transmit diversity gain equal to d. Galois ring includes the finite field F p k when a = 1 and the integer ring Z p a when k = 1. Our construction includes as a special case, the earlier construction by Lusina et.al. which is applicable only for RD codes over F p (p = 4s + 1) and transmit diversity gain d = n t .

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Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 87%

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 88%

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 99%

“…• STBCs with optimal rate-diversity tradeoff are constructed using codes over arbitrary Galois rings, which generalizes the map in [10].…”

Section: Space-time Codes Based On Rank-distance Codesmentioning

confidence: 99%

See 2 more Smart Citations

Optimal STBCs from codes over galois rings

Kiran

Rajan

2005 IEEE International Conference on Personal Wireless Communications, 2005. ICPWC 2005.

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“…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.…”

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confidence: 99%

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

Sripati

Shashidhar

Rajan

International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.

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-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...

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Random Network Coding and Matroids

Gadouleau¹

2012

Network Coding

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Network coding, a relatively new area of research, has evolved from the theoretical level to become a tool used to optimize the performance of communication networks-wired, cellular, ad hoc, etc. The idea consists of mixing "packets" of data together when routing them from source to destination. Since network coding increases the network performance, it becomes a tool to enhance the existing protocols and algorithms in a network or for applications such as peer-to-peer and TCP. This book delivers an understanding of network coding and provides a set of studies showing the improvements in security, capacity and performance of fixed and mobile networks. This is increasingly topical as industry is increasingly becoming more reliant upon and applying network coding in multiple applications. Many cases where network coding is used in routing, physical layer, security, flooding, error correction, optimization and relaying are given-all of which are key areas of interest. Network Coding is the ideal resource for university students studying coding, and researchers and practitioners in sectors of all industries where digital communication and its application needs to be correctly understood and implemented.

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Maximum rank distance codes as space~time codes

Cited by 97 publications

References 7 publications

Optimal STBCs from codes over galois rings

Optimal STBCs from codes over galois rings

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

Random Network Coding and Matroids

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