Description
A spherical code obtained from a binary code, \(q\)-ary code, or \(q\)-ary code over \(\mathbb{Z}_q\) via a component-wise mapping of each \(q\)-ary digit to a \(q\)th root of unity in a generalization of the antipodal mapping.
For example, for the case \(q=4\), one can map either the ring-based alphabet \(\mathbb{Z}_4 = \{0,1,2,3\}\) or the field-based alphabet \(\mathbb{F}_2^2 = \{00,01,10,11\}\) to the set \(\{1,\theta,\theta^2,\theta^3\}\) for some fourth root of unity \(\theta\).
Notes
See [12; Ch. 7] for more details.Cousins
- \(q\)-ary code— Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes.
- \(q\)-ary code over \(\mathbb{Z}_q\)— Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes over rings \(\mathbb{Z}_q\).
- Simplex spherical code— Simplex spherical codes for dimension \(n=(p-1)/2\) with \(p\) an odd prime admit a polyphase realization [12; Sec. 7.7].
- Biorthogonal spherical code— Biorthogonal spherical codes for dimension \(n=p\) with \(p\) an odd prime admit a polyphase realization [12; Sec. 7.7].
- \([6,3,4]_4\) Hexacode— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [13].
Member of code lists
Primary Hierarchy
Parents
A polyphase code can be thought of as a concatenation of a \(q\)-ary outer code with a PSK inner code.
Polyphase code
Children
A polyphase code can be thought of as a concatenation of a \(q\)-ary outer code with a PSK inner code. When the outer code is trivial, the construction reduces to a PSK code.
The polyphase mapping for \(q=2\) reduces to the antipodal mapping.
References
- [1]
- L.-H. Zetterberg, “A class of codes for polyphase signals on a bandlimited Gaussian channel”, IEEE Transactions on Information Theory 11, 385 (1965) DOI
- [2]
- L.-H. Zetterberg, “Detection of a class of coded and phase-modulated signals”, IEEE Transactions on Information Theory 12, 153 (1966) DOI
- [3]
- G. Einarsson, “Polyphase coding for a Gaussian channel(Polyphase coding for Gaussian channel, investigating PM signal transmission over channel disturbed by additive white Gaussian noise)”, Ericsson Technics 24(2), 75-130 (1968)
- [4]
- G. Einarsson. Performance of polyphase signals on a Gaussian channel. 1966
- [5]
- R. Ottoson, “Performance of phase- and amplitude-modulated signals on a Gaussian channel(Phase and amplitude modulated signals transmission over band limited channel disturbed by additive white Gaussian noise)”, Ericsson Technics 25(3), 153-198 (1969)
- [6]
- M. Nilsson, “Linear block codes over rings for phase shift keying”, PhD thesis, Linköping University, 1993
- [7]
- P. Piret, “Bounds for codes over the unit circle”, IEEE Transactions on Information Theory 32, 760 (1986) DOI
- [8]
- V. V. Ginzburg, “Multidimensional Signals for a Continuous Channel”, Problemy Peredachi Informatsii 20(1), 28–46 (1984); Problems of Information Transmission 20(1), 20–34 (1984)
- [9]
- S. L. Portnoi, “Characterizations of modulation and encoding systems as concatenated codes”, Problems of Information Transmission 21(3), 14-27 (1985)
- [10]
- V. V. Zyablov and S. L. Portnoi, “Modulation/Coding System for a Gaussian Channel”, Problemy Peredachi Informatsii 23(3), 18–26 (1987); Problems of Information Transmission 23(3), 187–193 (1987)
- [11]
- V. A. Zinoviev, S. N. Litsyn and S. L. Portnoi, Cascade codes in Euclidean space, Problems of Information Transmission, 25, (3), pp. 62-75, 1989
- [12]
- T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
- [13]
- V. V. Albert, private communication, 2024
Page edit log
- Victor V. Albert (2022-11-18) — most recent
Cite as:
“Polyphase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polyphase