# Gold code[1]

## Description

Member of the family of \([2^r-1, 2r ]\) cyclic binary linear codes characterized by the generator polynomial of degree \(r\) of two maximum-period sequences of period \(2^r-1\) with absolute cross-correlation \( \leq 2^{(r+2)/2}\). Gold codewords are generated using \(m\)-sequences \(x\) and \(y\), which are codewords of simplex codes with check polynomials of degree \(r\) [1].

## Encoding

Information bits are initialized in the shift registers of the two \(m\)-sequences \(x\) and \(y\).

## Decoding

General decoding is done by building a sparse parity check matrix, followed by applying an iterative message passing alogirithm. [2].

## Realizations

## Parent

## Cousins

- \([2^m-1,m,2^{m-1}]\) simplex code — Simplex codes are used to make gold codes. The dual of a Gold code is the interesection of the duals of the simplex codes used to construct it [5].
- Kasami code — Gold and Kasami codes are both constructed by picking a set of cyclically unrelated sequences of binary linear codes with low crosscorrelation [6,7].

## References

- [1]
- R. Gold, “Optimal binary sequences for spread spectrum multiplexing (Corresp.)”, IEEE Transactions on Information Theory 13, 619 (1967) DOI
- [2]
- O. W. Yeung and K. M. Chugg, “An Iterative Algorithm and Low Complexity Hardware Architecture for Fast Acquisition of Long PN Codes in UWB Systems”, Journal of VLSI signal processing systems for signal, image and video technology 43, 25 (2006) DOI
- [3]
- Mujtaba Hamid and Andy Miller, Gold Code Generators in Virtex Devices, (2000)
- [4]
- J. J. SPILKER Jr., “GPS Signal Structure and Performance Characteristics”, Navigation 25, 121 (1978) DOI
- [5]
- M. des Noes et al., “Iterative decoding of Gold sequences”, 2015 IEEE International Conference on Communications (ICC) (2015) DOI
- [6]
- D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences”, Proceedings of the IEEE 68, 593 (1980) DOI
- [7]
- T. Helleseth, C. Li, "Pseudo-Noise Sequences." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI

## Page edit log

- Victor V. Albert (2022-01-02) — most recent
- Khalil Guy (2022-01-02)

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/cyclic/gold.yml.