## 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.

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 \(GF(2)^2 = \{00,01,10,11\}\) to the set \(\{1,\theta,\theta^2,\theta^3\}\) for some fourth root of unity \(\theta\).

## Notes

See [12; Ch. 7] for more details.

## Parents

- Torus-layer spherical code (TLSC)
- Concatenated code — A polyphase code can be thought of as a concatenation of a \(q\)-ary outer code with a PSK inner code.

## Children

- Phase-shift keying (PSK) code — 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.
- Hypercube code
- Binary antipodal code

## Cousins

- Galois-field \(q\)-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].

## 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]
- Einarsson, Göran. "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 (1968): 75-130.
- [4]
- Einarsson, Göran. Performance of polyphase signals on a Gaussian channel. 1966.
- [5]
- Ottoson, Ragnar. "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 (1969): 153-198.
- [6]
- Nilsson, Magnus. "Linear block codes over rings for phase shift keying." Thesis no. 331, Linkoping 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”, Probl. Peredachi Inf., 20:1 (1984), 28–46; Problems Inform. Transmission, 20:1 (1984), 20–34
- [9]
- Portnoi, S. L. "Characterizations of modulation and encoding systems as concatenated codes." Probl. Inform. Transm. 21.3 (1985): 14-27.
- [10]
- V. V. Zyablov, S. L. Portnoi, “Modulation/Coding System for a Gaussian Channel”, Probl. Peredachi Inf., 23:3 (1987), 18–26; Problems Inform. Transmission, 23:3 (1987), 187–193
- [11]
- V.A. Zinoviev, S.N. Litsyn and S.L. Portnoi, Cascade codes in Euclidean space, Problems of Information Transmission, Vol. 25, No. 3, pp. 62-75, 1989.
- [12]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.

## 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