# Binary PSK (BPSK) code[1]

## Description

Also called a binary antipodal modulation, phase-reversal keying (PRK), or antipodal signaling. Encodes one bit of information into a constellation of antipodal points \(\pm\alpha\) for complex \(\alpha\). These points are typically associated with two phases of an electromagnetic signal.

## Rate

Achieve capacity of AGWN in the low signal-to-noise regime [2] (see also [1]). BPSK concatenated with quantum-classical polar codes achieves the Holevo capacity for the pure-loss channel [3].

## Realizations

Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [4] and V.23 [5].

## Parents

- Phase-shift keying (PSK) code — BPSK is also known as 2-PSK.
- Binary antipodal code — A single-bit binary code yields a spherical \((n,2,4)\) spherical code under the antipodal mapping, which is equivalent to the BPSK code for dimension \(n=2\).

## Cousins

- Pulse-amplitude modulation (PAM) code — BPSK for real \(\alpha\) is the simplest non-trivial PAM encoding.
- Linear binary code — Concatenating binary linear codes with BPSK yields a standard way of digitizing the analog AGWN channel [6; Ch. 29].
- Two-component cat code — BPSK (two-component cat) codes are used to transmit classical (quantum) information using (superpositions of) antipodal coherent states over classical (quantum) channels.
- Polar c-q code — BPSK concatenated with quantum-classical polar codes achieves the Holevo capacity for the pure-loss channel [3].
- Binary antipodal code — A binary antipodal code can be thought of as a concatenation of a binary outer code with a BPSK inner code.
- Binary balanced spherical code — A binary balanced spherical code can be thought of as a concatenation of a constant-weight binary outer code with a shifted and scaled BPSK-like inner code.
- BPSK c-q code — BPSK (BPSK c-q) codes are used to transmit classical information using antipodal coherent states over classical (quantum) channels.
- Hadamard BPSK c-q code — The Hadamard BPSK c-q code can be thought of as a concatenation of the Hadamard binary linear code with BPSK for the purposes of transmission of classical information over quantum channels.

## References

- [1]
- C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 623 (1948) DOI
- [2]
- Y. Wu and S. Verdu, “The impact of constellation cardinality on Gaussian channel capacity”, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (2010) DOI
- [3]
- S. Guha and M. M. Wilde, “Polar coding to achieve the Holevo capacity of a pure-loss optical channel”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0533 DOI
- [4]
- International Telecommunication Union-T, Recommendation V.21: 300 bits per second duplex modem standardized for use in the general switched telephone network, 1984
- [5]
- International Telecommunication Union-T, Recommendation V.23: 600/1200-baud modem standardized for use in the general switched telephone network, 1988
- [6]
- A. Lapidoth, A Foundation in Digital Communication (Cambridge University Press, 2017) DOI

## Page edit log

- Victor V. Albert (2022-11-07) — most recent

## Cite as:

“Binary PSK (BPSK) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bpsk