On the rank distance of cyclic codes

Sripati, U.; Rajan, B. Sundar

doi:10.1109/isit.2003.1228086

Cited by 10 publications

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“…Usual cyclic codes have been considered for the rank metric in [5,19] and a new construction, the so-called rank q-cyclic codes, was introduced in [7] for square matrices and has been generalized in [8] for other lengths. Independently, this notion has been generalized to skew or q r -cyclic codes in the work by Ulmer et al in [1,2,3], where r may be different from 1.…”

Section: Introductionmentioning

confidence: 99%

On the roots and minimum rank distance of skew cyclic codes

Martínez-Peñas

2016

Des. Codes Cryptogr.

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Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint-Wilson's shift and Hartmann-Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include all Hamming-metric cyclic codes, giving then a new relation between these codes and rank-metric skew cyclic codes

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Section: Introductionmentioning

confidence: 99%

On the roots and minimum rank distance of skew cyclic codes

Martínez-Peñas

2016

Des. Codes Cryptogr.

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“…The 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.…”

Section: Extended Summarymentioning

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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“…The proofs of the first two theorems are given in [6], [7] and that of the third is omitted due to space limitations.…”

Section: Characterisation Of Cyclic Codes For the Rank Metricmentioning

confidence: 99%

“…The case where rank q (C) = n − k + 1 has been studied in [4], [5], and are called Maximum Rank distance (MRD) codes. The rank properties of (n, k) cyclic codes over finite fields F q m , (n|q m − 1, (n, q) = 1) have been studied in [6], [7]. In these, exact expressions and tight bounds for the rank of the code have been derived by making use of the Discrete Fourier Transform (DFT) description of these codes.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we derive m × n STBCs having diversity equal to n, suitable for use over block fading channels, from n-length cyclic codes over F q m where n|q m − 1, m ≤ n by making use of the rank characterization for cyclic codes [6], [7]. This characterization has been performed by making use of the DFT domain description of cyclic codes [8].…”

Section: Introductionmentioning

confidence: 99%

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Full-diversity STBCs for block-fading channels from cyclic codes

Sripati

Rajan

Shashidhar

IEEE Global Telecommunications Conference, 2004. GLOBECOM '04.

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Abstract-Viewing an n-length vector over Fqm (the finite field of q m elements) 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. Using appropriate Discrete Fourier Transform (DFT), it is known that under some conditions, n-length cyclic codes over Fqm , (n|q m −1 and m ≤ n), have full-rank (= m). In this paper, using this result, we obtain designs for Full-diversity Space Time Block Codes (STBCs) suitable for block-fading channels from n length cyclic codes over Fqm . These STBCs are suitable for m transmit antennas over signal sets matched to Fq, where q = 2 or q is a prime of the form 4k + 1, (k = 1, 2, · · ·). We also present simulation results which illustrate the performance of a few of these STBCs and show that our codes perform better than the well known codes for block-fading channels.

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On the rank distance of cyclic codes

Cited by 10 publications

References 0 publications

On the roots and minimum rank distance of skew cyclic codes

On the roots and minimum rank distance of skew cyclic codes

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

Full-diversity STBCs for block-fading channels from cyclic codes

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