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doi: 10.3934/amc.2022003
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Two classes of cyclic extended double-error-correcting Goppa codes

1. 

Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211167, China

2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China

3. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

4. 

Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

Corresponding author: Qin Yue

Received  July 2021 Revised  November 2021 Early access February 2022

Fund Project: This paper was supported by National Natural Science Foundation of China (Nos. 61772015, 11961050, 12101277), Fundation of Nanjing Institute of Technology (No. CKJB202007), the Guangxi Natural Science Foundation(No. 2020GXNSFAA159053), Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-17-010)

Let $ \Bbb F_{2^m} $ be a finite extension of the field $ \Bbb F_2 $ and $ g(x) = x^2+\alpha x+1 $ a quadratic polynomial over $ \Bbb F_{2^m} $. In this paper, two classes of cyclic extended double-error-correcting Goppa codes are proposed. We obtain the following two classes of Goppa codes: (1) cyclic extended Goppa code with the irreducible polynomial $ g(x) $ and $ L = \Bbb F_{2^m}\cup \{\infty\} $; (2) cyclic extended Goppa code with the reducible polynomial $ g(x) $ and $ |L'| = 2^m-1 $. In addition, the parameters of above cyclic extended Goppa codes are given.

Citation: Yanyan Gao, Qin Yue, Xinmei Huang, Yun Yang. Two classes of cyclic extended double-error-correcting Goppa codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022003
References:
[1]

T. P. Berger, New classes of cyclic extended Goppa codes, IEEE Trans. Inf. Theory, 45 (1999), 1264-1266.  doi: 10.1109/18.761281.

[2]

T. P. Berger, On the cyclicity of Goppa codes, parity-check subcodes of Goppa codes, and extended Goppa codes, Finite Fields Appl., 6 (2000), 255-281.  doi: 10.1006/ffta.2000.0277.

[3]

T. P. Berger, Quasi-cyclic Goppa codes, IEEE Int. Symp. Inf. Theory, Sorrento, Italy, (2000), 195–195. doi: 10.1109/ISIT.2000.866493.

[4]

E. R. Berlekamp and O. Moreno, Extended doubule-error-correcting binary Goppa codes are cyclic, IEEE Trans. Inf. Theory, 19 (1973), 817-818.  doi: 10.1109/tit.1973.1055098.

[5]

S. V. Bezzateev and N. A. Shekhunova, Subclass of binary Goppa codes with minimal distance equal to design distance, IEEE Trans. Inf. Theory, 41 (1995), 554-555.  doi: 10.1109/18.370170.

[6]

S. V. Bezzateev and N. A. Shekhunova, Subclass of cyclic Goppa codes, IEEE Trans. Inf. Theory, 59 (2013), 7379-7385.  doi: 10.1109/TIT.2013.2278176.

[7]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.

[8]

Y. GaoQ. YueX. Huang and J. Zhang, Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to Quantum codes, IEEE Trans. Inf. Theorey, 67 (2021), 6619-6626.  doi: 10.1109/TIT.2021.3074526.

[9]

V. Goppa, A new class of linear error correcting codes, Problemy Peredači Informacii, 6 (1970), 207-212. 

[10]

M. JibrilS. BezzateevM. TomlinsonM. Grassl and M. Z. Ahmed, A generalized construction of extended Goppa codes, IEEE Trans. Inf. Theory, 60 (2014), 5296-5303.  doi: 10.1109/TIT.2014.2330814.

[11]

F. Levy-dit-Vehel and S. Litsyn, Parameters of Goppa codes revisited, IEEE Trans. Inf. Theory, 43 (1997), 1811-1819.  doi: 10.1109/18.641547.

[12]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.

[13]

F. Li and Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclic codes, Des. Codes Cryptogr., 86 (2018), 771-784.  doi: 10.1007/s10623-017-0356-2.

[14]

N. LiC. LiT. HellesethC. Ding and X. Tang, Optimal ternary cyclic codes with minimum distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.

[15]

Y. Liu and X. Cao, Four classes of optimal quinary cyclic codes, IEEE Commun. Let., 24 (2020), 1387-1390.  doi: 10.1109/LCOMM.2020.2983373.

[16]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977.

[17]

J. L. Massey, Reversible codes, Inf. Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.

[18]

H. Stichtenoth, Which extended Goppa codes are cyclic?, J. Combination Theory, 51 (1989), 205-220.  doi: 10.1016/0097-3165(89)90045-9.

[19]

M. TomlinsonS. V. BezzateevM. JibrilM. A. Ambroze and M. Z. Ahmed, Using the structure of subfields in the construction of Goppa codes and extended Goppa codes, IEEE Trans. Inf. Theory, 61 (2015), 3214-3224.  doi: 10.1109/TIT.2015.2419613.

[20]

K. K. Tzeng and C. Y. Yu, Characterization theorems for extending Goppa codes to cyclic codes, IEEE Trans. Inf. Theory, 25 (1979), 246-250.  doi: 10.1109/TIT.1979.1056016.

[21]

K. K. Tzeng and K. Zimmermann, On extending Goppa codes to cyclic codes, IEEE Trans. Inf. Theory, 21 (1975), 712-716.  doi: 10.1109/tit.1975.1055449.

[22]

J. H. van Lint, Introduction to Coding Theorey, Springer-Verlag New York Heidelberg Berlin, 1982.

[23]

A. L. Vishnevetskii, Cyclicity of extended Goppa codes, Probl. Peredachi Inf., 18 (1982), 14-18. 

[24]

L. Wang and G. Wu, Several classes of optimal ternary cyclic codes with minimal distance four, Finite Fields Appl., 40 (2016), 126-137.  doi: 10.1016/j.ffa.2016.03.007.

[25]

Y. Wu, Twisted Reed-Solomom codes with one-dimensional hull, IEEE Commun. Let., 25 (2021), 383-386. 

[26]

Y. WuQ. Yue and F. Li, Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings, Discrete Math., 341 (2018), 1755-1767.  doi: 10.1016/j.disc.2017.10.027.

show all references

References:
[1]

T. P. Berger, New classes of cyclic extended Goppa codes, IEEE Trans. Inf. Theory, 45 (1999), 1264-1266.  doi: 10.1109/18.761281.

[2]

T. P. Berger, On the cyclicity of Goppa codes, parity-check subcodes of Goppa codes, and extended Goppa codes, Finite Fields Appl., 6 (2000), 255-281.  doi: 10.1006/ffta.2000.0277.

[3]

T. P. Berger, Quasi-cyclic Goppa codes, IEEE Int. Symp. Inf. Theory, Sorrento, Italy, (2000), 195–195. doi: 10.1109/ISIT.2000.866493.

[4]

E. R. Berlekamp and O. Moreno, Extended doubule-error-correcting binary Goppa codes are cyclic, IEEE Trans. Inf. Theory, 19 (1973), 817-818.  doi: 10.1109/tit.1973.1055098.

[5]

S. V. Bezzateev and N. A. Shekhunova, Subclass of binary Goppa codes with minimal distance equal to design distance, IEEE Trans. Inf. Theory, 41 (1995), 554-555.  doi: 10.1109/18.370170.

[6]

S. V. Bezzateev and N. A. Shekhunova, Subclass of cyclic Goppa codes, IEEE Trans. Inf. Theory, 59 (2013), 7379-7385.  doi: 10.1109/TIT.2013.2278176.

[7]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.

[8]

Y. GaoQ. YueX. Huang and J. Zhang, Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to Quantum codes, IEEE Trans. Inf. Theorey, 67 (2021), 6619-6626.  doi: 10.1109/TIT.2021.3074526.

[9]

V. Goppa, A new class of linear error correcting codes, Problemy Peredači Informacii, 6 (1970), 207-212. 

[10]

M. JibrilS. BezzateevM. TomlinsonM. Grassl and M. Z. Ahmed, A generalized construction of extended Goppa codes, IEEE Trans. Inf. Theory, 60 (2014), 5296-5303.  doi: 10.1109/TIT.2014.2330814.

[11]

F. Levy-dit-Vehel and S. Litsyn, Parameters of Goppa codes revisited, IEEE Trans. Inf. Theory, 43 (1997), 1811-1819.  doi: 10.1109/18.641547.

[12]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.

[13]

F. Li and Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclic codes, Des. Codes Cryptogr., 86 (2018), 771-784.  doi: 10.1007/s10623-017-0356-2.

[14]

N. LiC. LiT. HellesethC. Ding and X. Tang, Optimal ternary cyclic codes with minimum distance four and five, Finite Fields Appl., 30 (2014), 100-120.  doi: 10.1016/j.ffa.2014.06.001.

[15]

Y. Liu and X. Cao, Four classes of optimal quinary cyclic codes, IEEE Commun. Let., 24 (2020), 1387-1390.  doi: 10.1109/LCOMM.2020.2983373.

[16]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977.

[17]

J. L. Massey, Reversible codes, Inf. Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.

[18]

H. Stichtenoth, Which extended Goppa codes are cyclic?, J. Combination Theory, 51 (1989), 205-220.  doi: 10.1016/0097-3165(89)90045-9.

[19]

M. TomlinsonS. V. BezzateevM. JibrilM. A. Ambroze and M. Z. Ahmed, Using the structure of subfields in the construction of Goppa codes and extended Goppa codes, IEEE Trans. Inf. Theory, 61 (2015), 3214-3224.  doi: 10.1109/TIT.2015.2419613.

[20]

K. K. Tzeng and C. Y. Yu, Characterization theorems for extending Goppa codes to cyclic codes, IEEE Trans. Inf. Theory, 25 (1979), 246-250.  doi: 10.1109/TIT.1979.1056016.

[21]

K. K. Tzeng and K. Zimmermann, On extending Goppa codes to cyclic codes, IEEE Trans. Inf. Theory, 21 (1975), 712-716.  doi: 10.1109/tit.1975.1055449.

[22]

J. H. van Lint, Introduction to Coding Theorey, Springer-Verlag New York Heidelberg Berlin, 1982.

[23]

A. L. Vishnevetskii, Cyclicity of extended Goppa codes, Probl. Peredachi Inf., 18 (1982), 14-18. 

[24]

L. Wang and G. Wu, Several classes of optimal ternary cyclic codes with minimal distance four, Finite Fields Appl., 40 (2016), 126-137.  doi: 10.1016/j.ffa.2016.03.007.

[25]

Y. Wu, Twisted Reed-Solomom codes with one-dimensional hull, IEEE Commun. Let., 25 (2021), 383-386. 

[26]

Y. WuQ. Yue and F. Li, Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings, Discrete Math., 341 (2018), 1755-1767.  doi: 10.1016/j.disc.2017.10.027.

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