The Hamming and Golay Number-Theoretic Transforms

Paschoal, A. J. A.; Souza, Ricardo M. Campello de; Oliveira, H. M. de

doi:10.14209/sbrt.2018.1

Cited by 2 publications

(6 citation statements)

References 12 publications

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“…The special case p = 2 and N = 7 is suitably linked to the Hamming NTT [17]. Based on the binary Hamming code H (7,4,3), we get the 7 × 7 binary Hamming NTT, whose transformation matrix is [18]…”

Section: Geometrical Representation Of Binary Sequencesmentioning

confidence: 99%

“…Based on the Fourier NTT and the Hartley NTT, the Fourier and Hartley codes were introduced [3,7]. Conversely, popular error-correcting codes [13], such as the Hamming [9] and Golay codes [14], inspired the introduction of the Hamming number-theoretic transform (HamNTT) [17] and the Golay number-theoretic transform (GNTT) [17] which extend the theory introduced in [4,18]. In fact, an isomorphism between linear codes and transforms was identified in [17].…”

Section: Introductionmentioning

confidence: 99%

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Geometrical Representation for Number-theoretic Transforms

Oliveira¹,

Cintra²

2021

Preprint

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This short note introduces a geometric representation for binary (or ternary) sequences. The proposed representation is linked to multivariate data plotting according to the radar chart. As an illustrative example, the binary Hamming transform recently proposed is geometrically interpreted. It is shown that codewords of standard Hamming code H (N = 7, k = 4, d = 3) are invariant vectors under the Hamming transform. These invariant are eigenvectors of the binary Hamming transform. The images are always inscribed in a regular polygon of unity side, resembling triangular rose petals and/or "thorns". A geometric representation of the ternary Golay transform, based on the extended Golay G (N = 12, k = 6, d = 6) code over GF(3) is also showed. This approach is offered as an alternative representation of finite-length sequences over finite prime fields.

show abstract

Section: Geometrical Representation Of Binary Sequencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Geometrical Representation for Number-theoretic Transforms

Oliveira¹,

Cintra²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The special case p = 2 and N = 7 is suitably linked to the Hamming NTT [16]. Based on the binary Hamming code H(7, 4, 3), we get the 7 × 7 binary Hamming NTT, whose transformation matrix is [18]…”

Section: Geometrical Representation Of Binary Sequencesmentioning

confidence: 99%

“…Thus the Golay NTT requires only additions in order to be computed and it is applicable to any sequence of the {GF(3)} 12 -space. Illustrative examples of the effect of the Golay numbertheoretic transform [16] on ternary vectors of length 12 are shown in Figures 5 and 6. Note that complex symbols are always vertices of one of the two dodecagons.…”

Section: Geometrical Representation Of the Ternary Golay Transformmentioning

confidence: 99%

“…Based on the Fourier NTT and the Hartley NTT, the Fourier and Hartley codes were introduced [11], [12]. Conversely, popular error-correcting codes [13], such as the Hamming [14] and Golay codes [15], inspired the introduction of the Hamming number-theoretic transform (HamNTT) [16] and the Golay number-theoretic transform (GNTT) [16] which extend the theory introduced in [17], [18]. In fact, an isomorphism between linear codes and transforms was identified in [16].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometrical Representation for Number-theoretic Transforms

Oliveira¹,

Cintra²

2020

Anais De XXXVIII Simpósio Brasileiro De Telecomunicações E Processamento De Sinais

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This short note introduces a geometric representation for binary (or ternary) sequences. The proposed representation is linked to multivariate data plotting according to the radar chart. As an illustrative example, the binary Hamming transform recently proposed is geometrically interpreted. It is shown that codewords of standard Hamming code H(N = 7, k = 4, d = 3) are invariant vectors under the Hamming transform. These invariant are eigenvectors of the binary Hamming transform. The images are always inscribed in a regular polygon of unity side, resembling triangular rose petals and/or "thorns". A geometric representation of the ternary Golay transform, based on the extended Golay G(N = 12, k = 6, d = 6) code over GF(3) is also showed. This approach is offered as an alternative representation of finite-length sequences over finite prime fields.

show abstract

The Hamming and Golay Number-Theoretic Transforms

Cited by 2 publications

References 12 publications

Geometrical Representation for Number-theoretic Transforms

Geometrical Representation for Number-theoretic Transforms

Geometrical Representation for Number-theoretic Transforms

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