Ensemble of Gold’s codes generation for direct-sequence spread spectrum

Аuthors

Kuznetsov V. S.^1*, Shevchenko I. V.^1**, Volkov A. S.^2***, Solodkov A. V.^2****

1. National Research University of Electronic Technology, 1, sq. Shokina, Moscow, Zelenograd, 124498, Russia
2. National Research University of Electronic Technology, Bld. 1, Shokin Square, Zelenograd, Moscow, Russia, 124498

*e-mail: vitaliy_kuznetsov@hotmail.com
**e-mail: tcs@mee.ru
***e-mail: leshvol@mail.ru
****e-mail: solodkov_aw@mail.ru

Abstract

The article considers generation of m-sequences and Gold’s codes employed in spectrum spreading systems. The goal of the article consists in ensembles systematization and search for all pairs of polynomials, forming Gold’s codes.

Gold’s codes have a low level of cross-correlation between the sequences in the ensemble, which allows employ them for users’ separation in communication systems. Maximum number of users in the system depends on the number of sequences in the code ensemble.

The classical method of e-sequences and Gold’s codes generation is representation of generator in the form of linear-feedback shift register (LFSR), since this approach ensures the simplicity of algorithm implementation to a digital unit. Employing the classical generation method (based on the generator representation in the form of LFSR) becomes non-optimal while generating ensembles of long sequences (N > 2¹⁰ – 1). The generation time of all Gold’s code pairs grows significantly due to the number of all possible combinations increasing while sorting pairs of sequences.

The Gold’s code generation method using the Berlekamp-Massey algorithm and decimations defined for the codes under consideration generation was selected. Based on this algorithm the authors developed the program for searching the preferable pairs returning the Gold’s code under addition modulo two. The main goal of the program is generation of all volume of the Gold’s code family. The obtained results were checked by evaluation of the value and number of cross-correlation peaks of the two tested sequences. The obtained results can be implemented in the systems employing direct spectrum spreading, such as CDMA.

The paper presents the number of binary polynomials and the list of the number of unique polynomial pairs for the code length up to m ≤ 16, forming Gold’s codes. The full list of primitive polynomials for the code length of N = 2¹⁰ – 1 is presented. The example of preferable pairs of primitive polynomials of m-th degree (m = 5) is given. The number of pairs polynomials for generating Gold-like codes for the degrees of m = 8 and m = 12 of code seeds is given.

The obtained results can be implemented while developing communication systems with users’ separation for obtaining overall Gold’s ensemble and studying the properties of the obtained sequences and their derivatives.

Keywords:

m-sequences, Gold’s codes, preferred pairs, decimation coefficients, binary polynomials, DSSS, linear-feedback shift register

References

1. Pestryakov V.B. Shumopodobnye signaly v sistemakh peredachi informatsii (Noise-like signals in data transfer systems), Moscow, Sovetskoe radio, 1973, 424 p.

2. Ipatov V.P. Shirokopolosnye sistemy i kodovoe razdelenie signalov (Wideband and code division systems), Moscow, Mir svyazi, 2007, 488 p.

3. Varakin L.E. Sistemy svyazi s shumopodobnymi signalami (Data transfer systems with noise-like signals), Moscow, Radio i svyaz’, 1985, 384 c.

4. Barinov V.V., Lebedev M.V., Kuznetsov B.C. Elektrosvyaz’, 2006, no. 3. pp. 38-39.

5. Solomon W. Golomb, and Guang Gong. Signal Design for Good Correlation, Cambridge, Cambridge University Press, 2005, 458 p.

6. Pingzhi Fan and Mike Darnell. Sequence design for communications applications, London, Research Studie, 1996, 516 p.

7. Simon K. Marvin. Spread spectrum communications handbook, New York, McGraw-Hill, 1994, 1228 p.

8. Zierler Neal and John Brillhart. On primitive trinomials (mod 2). Information and Control, 1968, no. 13, pp. 541-554.

9. Stahnke Wayne. Primitive binary polynomials. Mathematics of Computation, 1973, 27(124), pp. 977-980.

10. Živković Miodrag. Generation of primitive binary polynomials. Filomat, 1995, vol. 9, no.3, pp. 961-965.

11. Borodin V. V., Petrakov A.M., Shevcov V.A. Trudy MAI, 2015, no. 81, available at: http://trudymai.ru/eng/published.php?ID=57894

12. Bogdanov A.S., Shevcov V.A. Trudy MAI, 2010, no. 40, available at: http://trudymai.ru/eng/published.php?ID=22874

13. Bogdanov A.S., Shevcov V.A. Trudy MAI, 2015, no. 84, available at: http://trudymai.ru/eng/published.php?ID=63136

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