American Institute of Mathematical Sciences

Advanced
November  2021, 15(4): 663-676. doi: 10.3934/amc.2020088

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

Rong Wang 1,2, , Xiaoni Du 1,, and  Cuiling Fan 3,

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
2. 

Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
3. 

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610000, China

* Corresponding author: Xiaoni Du

Received  December 2019 Revised  February 2020 Published  November 2021 Early access  June 2020

Fund Project: The second author is supported by NSFC grant No. 61772022. The third author is supported by NSFC grant No. 11971395

Combinatorial $ t $-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $ t $-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $ t $-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $ t $-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of $ 2 $-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.

Keywords: Linear codes, cyclic codes, affine-invariant codes, exponential sums, weight distributions, 2-designs.
Mathematics Subject Classification: Primary: 05B05, 94B05, 11T23, 11T71.
Citation: Rong Wang, Xiaoni Du, Cuiling Fan. Infinite families of 2-designs from a class of non-binary Kasami cyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 663-676. doi: 10.3934/amc.2020088
References:
[1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

E. F. Assmus Jr. and H. F. Mattson Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-152.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.

[4]

M. Antweiler and L. Bömer, Complex sequences over GF$ {(p^M)} $ with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, 38 (1992), 120-130.  doi: 10.1109/18.108256.

[5]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. II. Encyclopedia of Mathematics and its Applications, Vol. 78, 2nd edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139507660.003.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[8]

C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[9]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[10]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of linear codes, preprint, arXiv: 1903.07459.

[11]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros, preprint, arXiv: 1903.08153.

[12]

X. Du, R. Wang and C. Fan, Infinite families of $2$-designs from a class of cyclic codes, J. Comb. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[13]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110.  doi: 10.1016/j.ffa.2004.06.002.

[14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[15]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, $2^nd$ edition, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.

[16]

T. Kasami, S. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 (1967), 475-496.  doi: 10.1016/S0019-9958(67)90691-2.

[17]

J. Luo, Y. Tang, and H. Wang, Exponential sums, cycle codees and sequences: the odd characteristic Kasami case, preprint, arXiv: 0902.4508v1 [cs.IT].

[18]

R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997.

[19]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[20]

C. Reid and A. Rosa, Steiner systems $ {S} (2, 4, v) $-a survey, The Electronic Journal of Combinatorics, 18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39.

[21]

J. Serrin, C. J. Colbourn and R. Mathon, Steiner systems, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, (2006), 128–135. https://www.researchgate.net/publication/329786723.

[22]

V. D. Tonchev, Codes and designs, in Handbook of Coding Theory, Vol. I, II North-Holland, Amsterdam, (1998), 1229–1267. https://www.researchgate.net/publication/268549395.

[23]

V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.

[24]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

show all references

References:
[1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

E. F. Assmus Jr. and H. F. Mattson Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-152.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.

[4]

M. Antweiler and L. Bömer, Complex sequences over GF$ {(p^M)} $ with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, 38 (1992), 120-130.  doi: 10.1109/18.108256.

[5]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. II. Encyclopedia of Mathematics and its Applications, Vol. 78, 2nd edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139507660.003.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[8]

C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[9]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[10]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of linear codes, preprint, arXiv: 1903.07459.

[11]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros, preprint, arXiv: 1903.08153.

[12]

X. Du, R. Wang and C. Fan, Infinite families of $2$-designs from a class of cyclic codes, J. Comb. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[13]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110.  doi: 10.1016/j.ffa.2004.06.002.

[14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[15]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, $2^nd$ edition, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.

[16]

T. Kasami, S. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 (1967), 475-496.  doi: 10.1016/S0019-9958(67)90691-2.

[17]

J. Luo, Y. Tang, and H. Wang, Exponential sums, cycle codees and sequences: the odd characteristic Kasami case, preprint, arXiv: 0902.4508v1 [cs.IT].

[18]

R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997.

[19]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[20]

C. Reid and A. Rosa, Steiner systems $ {S} (2, 4, v) $-a survey, The Electronic Journal of Combinatorics, 18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39.

[21]

J. Serrin, C. J. Colbourn and R. Mathon, Steiner systems, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, (2006), 128–135. https://www.researchgate.net/publication/329786723.

[22]

V. D. Tonchev, Codes and designs, in Handbook of Coding Theory, Vol. I, II North-Holland, Amsterdam, (1998), 1229–1267. https://www.researchgate.net/publication/268549395.

[23]

V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.

[24]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

Table 1.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd
Weight Multiplicity
$ 0 $ $ 1 $
$ (p-1)(p^{m-1}-p^{s-1}) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{m-1}(p-1)+p^{s-1} $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ \frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)-p^{s-1} $ $ \frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d} $
$ -p^{m-2d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table Options

Download as excel

Weight Multiplicity
$ 0 $ $ 1 $
$ (p-1)(p^{m-1}-p^{s-1}) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{m-1}(p-1)+p^{s-1} $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ \frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)-p^{s-1} $ $ \frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d} $
$ -p^{m-2d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table 2.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s-1) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p-1)(p^s+1) $ $ p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1) $ $ \frac{1}{2}p^{3s-2d}(p^m-1) $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1) $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d} $
$ +1)(p^m-1) $
$ p^m $ $ p-1 $
Table Options

Download as excel

Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s-1) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p-1)(p^s+1) $ $ p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1) $ $ \frac{1}{2}p^{3s-2d}(p^m-1) $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1) $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d} $
$ +1)(p^m-1) $
$ p^m $ $ p-1 $
Table 3.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s+1) $ $ \frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ \frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ \frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ \frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ \frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ \frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d} $
$ +p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d} $
$ -p^{m-4d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table Options

Download as excel

Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s+1) $ $ \frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ \frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ \frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ \frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ \frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ \frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d} $
$ +p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d} $
$ -p^{m-4d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table 4.  The value of $ T(a,b,c,h) $ when $ d' = d $ is odd ($ j\in \mathbb{F}_p $)
Value Corresponding Condition
$ p^{m-1}+\varepsilon p^{s-1}(p-1) $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0 $
$ p^{m-1}-\varepsilon p^{s-1} $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0 $
$ 0 $ $ S(a,b,c)= p^m\; \mathrm{and}\; h\neq0 $
$ p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon $
$ p^{m-1}-p^{s+d-1}(p-1) $ $ S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0 $
$ p^{m-1}+p^{s+d-1} $ $ S(a,b,c) $ $ =-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0 $
$ p^{m-1} $ $ S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and} $
$ h+j = 0 $
$ p^m $ $ S(a,b,c)=p^m\; \mathrm{and}\; h = 0 $
Table Options

Download as excel

Value Corresponding Condition
$ p^{m-1}+\varepsilon p^{s-1}(p-1) $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0 $
$ p^{m-1}-\varepsilon p^{s-1} $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0 $
$ 0 $ $ S(a,b,c)= p^m\; \mathrm{and}\; h\neq0 $
$ p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon $
$ p^{m-1}-p^{s+d-1}(p-1) $ $ S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0 $
$ p^{m-1}+p^{s+d-1} $ $ S(a,b,c) $ $ =-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0 $
$ p^{m-1} $ $ S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and} $
$ h+j = 0 $
$ p^m $ $ S(a,b,c)=p^m\; \mathrm{and}\; h = 0 $
Table 5.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd
Value Multiplicity
$ (p-1)(p^{m-1}-p^{s-1}) $ $ M_1+(p-1)M_3 $
$ p^{m-1}(p-1)+p^{s-1} $ $ (p-1)M_1+(p-1)^2M_3 $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ M_2+(p-1)M_4 $
$ p^{m-1}(p-1)-p^{s-1} $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ (p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1}) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1}) $
$ p^m $ $ p-1 $
Table Options

Download as excel

Value Multiplicity
$ (p-1)(p^{m-1}-p^{s-1}) $ $ M_1+(p-1)M_3 $
$ p^{m-1}(p-1)+p^{s-1} $ $ (p-1)M_1+(p-1)^2M_3 $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ M_2+(p-1)M_4 $
$ p^{m-1}(p-1)-p^{s-1} $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ (p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1}) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1}) $
$ p^m $ $ p-1 $
Table 6.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even
Value Multiplicity
$ p^{s-1}(p-1)(p^s-1) $ $ M_1+(p-1)M_3 $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ (p-1)M_1+(p-1)^2M_3 $
$ p^{s-1}(p-1)(p^s+1) $ $ M_2+(p-1)M_4 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1) $ $ (p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1} $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1) $ $ M_{5,\mp1}+(p-1)M_{6,\mp1} $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9 $
$ p^m $ $ p-1 $
Table Options

Download as excel

Value Multiplicity
$ p^{s-1}(p-1)(p^s-1) $ $ M_1+(p-1)M_3 $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ (p-1)M_1+(p-1)^2M_3 $
$ p^{s-1}(p-1)(p^s+1) $ $ M_2+(p-1)M_4 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1) $ $ (p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1} $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1) $ $ M_{5,\mp1}+(p-1)M_{6,\mp1} $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9 $
$ p^m $ $ p-1 $
Table 7.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $
Value Multiplicity
$ p^{s-1}(p-1)(p^s+1) $ $ M_1+(p-1)M_2 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_1+(p-1)^2M_2 $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ M_3+(p-1)M_4 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ (p-1)M_3+(p-1)^2M_4 $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ M_5+(p-1)M_6 $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ (p-1)M_5+(p-1)^2M_6 $
$ p^{m-1}(p-1) $ $ pM_7 $
$ p^m $ $ p-1 $
Table Options

Download as excel

Value Multiplicity
$ p^{s-1}(p-1)(p^s+1) $ $ M_1+(p-1)M_2 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_1+(p-1)^2M_2 $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ M_3+(p-1)M_4 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ (p-1)M_3+(p-1)^2M_4 $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ M_5+(p-1)M_6 $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ (p-1)M_5+(p-1)^2M_6 $
$ p^{m-1}(p-1) $ $ pM_7 $
$ p^m $ $ p-1 $
Table 8.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is odd
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}} $ $ M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table Options

Download as excel

Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}} $ $ M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table 9.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is even
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon p^{s+\frac{d}{2}} $ $ M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1) $
$ \varepsilon p^{s+\frac{d}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table Options

Download as excel

Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon p^{s+\frac{d}{2}} $ $ M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1) $
$ \varepsilon p^{s+\frac{d}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table 10.  The value distribution of $ S(a,b,c) $ when $ d' = 2d $
Value Multiplicity
$ -p^s $ $ M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)} $
$ -\zeta^j_pp^s $ $ M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d} $ $ M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d}\zeta^j_p $ $ M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ -p^{s+2d} $ $ M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ -p^{s+2d}\zeta^j_p $ $ M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ 0 $ $ M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}- $
$ 2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table Options

Download as excel

Value Multiplicity
$ -p^s $ $ M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)} $
$ -\zeta^j_pp^s $ $ M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d} $ $ M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d}\zeta^j_p $ $ M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ -p^{s+2d} $ $ M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ -p^{s+2d}\zeta^j_p $ $ M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ 0 $ $ M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}- $
$ 2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1) $
$ p^m $ $ 1 $
[1]

Yan Liu, Xiwang Cao. Infinite families of 2-designs from a class of affine-invariant codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022011
[2]

Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang. Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros. Advances in Mathematics of Communications, 2022, 16 (1) : 157-168. doi: 10.3934/amc.2020106
[3]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161
[4]

Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
[5]

Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277
[6]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[7]

Tonghui Zhang, Hong Lu, Shudi Yang. Two-weight and three-weight linear codes constructed from Weil sums. Mathematical Foundations of Computing, 2022, 5 (2) : 129-144. doi: 10.3934/mfc.2021041
[8]

Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341
[9]

Jamshid Moori, Amin Saeidi. Some designs and codes invariant under the Tits group. Advances in Mathematics of Communications, 2017, 11 (1) : 77-82. doi: 10.3934/amc.2017003
[10]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[11]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195
[12]

Sara D. Cardell, Joan-Josep Climent, Daniel Panario, Brett Stevens. A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes. Advances in Mathematics of Communications, 2020, 14 (3) : 437-453. doi: 10.3934/amc.2020047
[13]

Toshiharu Sawashima, Tatsuya Maruta. Nonexistence of some ternary linear codes with minimum weight -2 modulo 9. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021052
[14]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[15]

Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059
[16]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
[17]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[18]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002
[19]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008
[20]

Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013

2021 Impact Factor: 1.015

Tools

Metrics

  • PDF downloads (253)
  • HTML views (637)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy