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doi: 10.3934/amc.2022006
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Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", Viale Lincoln, 5, I-81100 Caserta, Italy

*Corresponding author: Ferdinando Zullo

Received  July 2021 Revised  December 2021 Early access February 2022

Fund Project: We want to thank the referees for their careful reading and for their valuable comments. The research was supported by the project "VALERE: VAnviteLli pEr la RicErca" of the University of Campania "Luigi Vanvitelli" and was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM)

In this paper we consider two pointsets in $ \mathrm{PG}(2,q^n) $ arising from a linear set $ L $ of rank $ n $ contained in a line of $ \mathrm{PG}(2,q^n) $: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set $ L $. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing $ L $ to be an $ {\mathbb F}_{q} $-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the $ \Gamma\mathrm{L} $-class of $ L $ and the number of inequivalent codes we can construct starting from it.

Citation: Vito Napolitano, Olga Polverino, Paolo Santonastaso, Ferdinando Zullo. Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022006
References:
[1]

G. N. Alfarano, M. Borello, A. Neri and A. Ravagnani, Linear cutting blocking sets and minimal codes in the rank metric, arXiv: 2106.12465.

[2]

S. Ball, The number of directions determined by a function over finite field, J. Combin. Theory Ser. A, 104 (2003), 341-350.  doi: 10.1016/j.jcta.2003.09.006.

[3]

S. BallA. BlokhuisA. E. BrouwerL. Storme and T. Szőnyi, On the number of slopes of the graph of a function defined on a finite field, J. Combin. Theory Ser. A, 86 (1999), 187-196.  doi: 10.1006/jcta.1998.2915.

[4]

D. BartoliB. Csajbók and M. Montanucci, On a conjecture about maximum scattered subspaces of $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$, Linear Algebra Appl., 631 (2021), 111-135.  doi: 10.1016/j.laa.2021.08.023.

[5]

D. BartoliC. Zanella and F. Zullo, A new family of maximum scattered linear sets in ${\text{PG}}(1, q^6)$, Ars Math. Contemp., 19 (2020), 125-145.  doi: 10.26493/1855-3974.2137.7fa.

[6]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in ${\text{PG}}(n, q)$, Geom. Dedicata, 81 (2000), 231-243.  doi: 10.1023/A:1005283806897.

[7]

G. Bonoli and O. Polverino, $\mathbb{F}_q$-linear blocking sets in ${\text{PG}}(2, q^4)$, Innov. Incidence Geom., 2 (2005), 35-56.  doi: 10.2140/iig.2005.2.35.

[8]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[9]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.

[10]

B. CsajbókG. Marino and O. Polverino, Classes and equivalence of linear sets in ${\text{PG}}(1, q^n)$, J. Combin. Theory Ser. A, 157 (2018), 402-426.  doi: 10.1016/j.jcta.2018.03.007.

[11]

B. CsajbókG. Marino and O. Polverino, A Carlitz type results for linearized polynomials, Ars Math. Contemp., 16 (2019), 585-608.  doi: 10.26493/1855-3974.1651.e79.

[12]

B. CsajbókG. MarinoO. Polverino and C. Zanella, A new family of MRD-codes, Linear Algebra Appl., 548 (2018), 203-220.  doi: 10.1016/j.laa.2018.02.027.

[13]

B. CsajbókG. Marino and F. Zullo, New maximum scattered linear sets of the projective line, Finite Fields Appl., 54 (2018), 133-150.  doi: 10.1016/j.ffa.2018.08.001.

[14]

J. De Beule and G. Van de Voorde, The minimum size of a linear set, J. Combin. Theory Ser. A, 164 (2019), 109-124.  doi: 10.1016/j.jcta.2018.12.008.

[15]

M. De Boeck and G. Van de Voorde, A linear set view on KM-arcs, J. Algebraic Combin., 44 (2016), 131-164.  doi: 10.1007/s10801-015-0661-7.

[16]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoret. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[17]

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.

[18]

N. Durante, On sets with few intersection numbers in finite projective and affine spaces, Electron. J. Combin., 21 (2014), 1-18. 

[19]

A. Gács and Z. Weiner, On $(q+t)$-arcs of type $(0, 2, t)$, Des. Codes Cryptogr., 29 (2003), 131-139.  doi: 10.1023/A:1024152424893.

[20]

G. Korchmáros and F. Mazzocca, On $(q+t)$-arcs of type $(0, 2, t)$ in a desarguesian plane of order $q$, Math. Proc. Camb. Philos. Soc., 108 (1990), 445-459.  doi: 10.1017/S0305004100069346.

[21]

I. N. Landjev, Linear codes over finite fields and finite projective geometries, Discrete Math., 213 (2000), 211-244.  doi: 10.1016/S0012-365X(99)00183-1.

[22]

M. LavrauwG. MarinoO. Polverino and R. Trombetti, Solution to an isotopism question concerning rank $2$ semifields, J. Combin. Des., 23 (2015), 60-77.  doi: 10.1002/jcd.21382.

[23]

M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, Contemp. Math, Amer. Math. Soc., Providence, RI, 632 (2015), 271–293. doi: 10.1090/conm/632/12633.

[24]

G. Longobardi, G. Marino, R. Trombetti and Y. Zhou, A large family of maximum scattered linear sets of ${\text{PG}}(1, q^n)$ and their associated MRD codes, arXiv: 2102.08287.

[25]

G. Longobardi and C. Zanella, Linear sets and MRD-codes arising from a class of scattered linearized polynomials, J. Algebr. Combin., 53 (2021), 639-661.  doi: 10.1007/s10801-020-01011-9.

[26]

G. Lunardon and O. Polverino, Blocking sets of size $q^t + q^{t-1} + 1$, J. Combin. Theory Ser. A, 90 (2000), 148-158.  doi: 10.1006/jcta.1999.3022.

[27]

G. MarinoM. Montanucci and F. Zullo, MRD-codes arising from the trinomial $x^q + x^{q^3}+ cx^{q^5} \in {\mathbb F}_{q^6}[x]$, Linear Algebra Appl., 591 (2020), 99-114.  doi: 10.1016/j.laa.2020.01.004.

[28]

V. Napolitano, O. Polverino, P. Santonastaso and F. Zullo, Linear sets on the projective line with complementary weights, arXiv: 2107.10641.

[29]

V. Napolitano, O. Polverino, P. Santonastaso and F. Zullo, Classifications and constructions of minimum size linear sets, submitted.

[30]

V. Napolitano and F. Zullo, Codes with few weights arising from linear sets, Adv. Math. Commun.. doi: 10.3934/amc.2020129.

[31]

A. Neri, P. Santonastaso and F. Zullo, Extending two families of maximum rank distance codes, arXiv: 2104.07602.

[32]

O. Polverino, Linear sets in finite projective spaces, Discrete Math., 310 (2010), 3096-3107.  doi: 10.1016/j.disc.2009.04.007.

[33]

O. Polverino and F. Zullo, On the number of roots of some linearized polynomials, Linear Algebra Appl., 601 (2020), 189-218.  doi: 10.1016/j.laa.2020.05.009.

[34]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.

[35]

M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Des. Codes Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7.

[36]

M. A. Tsfasman, S. G. Vlǎduţ and D. Nogin, Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, 139. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/139.

[37]

C. Zanella and F. Zullo, Vertex properties of maximum scattered linear sets of $ {\text{PG}}(1, q^n)$, Discrete Math., 343 (2020), 111800, 14 pp. doi: 10.1016/j.disc.2019.111800.

[38]

G. Zini and F. Zullo, Scattered subspaces and related codes, Des. Codes Cryptogr., 89 (2021), 1853-1873.  doi: 10.1007/s10623-021-00891-7.

show all references

References:
[1]

G. N. Alfarano, M. Borello, A. Neri and A. Ravagnani, Linear cutting blocking sets and minimal codes in the rank metric, arXiv: 2106.12465.

[2]

S. Ball, The number of directions determined by a function over finite field, J. Combin. Theory Ser. A, 104 (2003), 341-350.  doi: 10.1016/j.jcta.2003.09.006.

[3]

S. BallA. BlokhuisA. E. BrouwerL. Storme and T. Szőnyi, On the number of slopes of the graph of a function defined on a finite field, J. Combin. Theory Ser. A, 86 (1999), 187-196.  doi: 10.1006/jcta.1998.2915.

[4]

D. BartoliB. Csajbók and M. Montanucci, On a conjecture about maximum scattered subspaces of $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$, Linear Algebra Appl., 631 (2021), 111-135.  doi: 10.1016/j.laa.2021.08.023.

[5]

D. BartoliC. Zanella and F. Zullo, A new family of maximum scattered linear sets in ${\text{PG}}(1, q^6)$, Ars Math. Contemp., 19 (2020), 125-145.  doi: 10.26493/1855-3974.2137.7fa.

[6]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in ${\text{PG}}(n, q)$, Geom. Dedicata, 81 (2000), 231-243.  doi: 10.1023/A:1005283806897.

[7]

G. Bonoli and O. Polverino, $\mathbb{F}_q$-linear blocking sets in ${\text{PG}}(2, q^4)$, Innov. Incidence Geom., 2 (2005), 35-56.  doi: 10.2140/iig.2005.2.35.

[8]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[9]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.

[10]

B. CsajbókG. Marino and O. Polverino, Classes and equivalence of linear sets in ${\text{PG}}(1, q^n)$, J. Combin. Theory Ser. A, 157 (2018), 402-426.  doi: 10.1016/j.jcta.2018.03.007.

[11]

B. CsajbókG. Marino and O. Polverino, A Carlitz type results for linearized polynomials, Ars Math. Contemp., 16 (2019), 585-608.  doi: 10.26493/1855-3974.1651.e79.

[12]

B. CsajbókG. MarinoO. Polverino and C. Zanella, A new family of MRD-codes, Linear Algebra Appl., 548 (2018), 203-220.  doi: 10.1016/j.laa.2018.02.027.

[13]

B. CsajbókG. Marino and F. Zullo, New maximum scattered linear sets of the projective line, Finite Fields Appl., 54 (2018), 133-150.  doi: 10.1016/j.ffa.2018.08.001.

[14]

J. De Beule and G. Van de Voorde, The minimum size of a linear set, J. Combin. Theory Ser. A, 164 (2019), 109-124.  doi: 10.1016/j.jcta.2018.12.008.

[15]

M. De Boeck and G. Van de Voorde, A linear set view on KM-arcs, J. Algebraic Combin., 44 (2016), 131-164.  doi: 10.1007/s10801-015-0661-7.

[16]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoret. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[17]

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.

[18]

N. Durante, On sets with few intersection numbers in finite projective and affine spaces, Electron. J. Combin., 21 (2014), 1-18. 

[19]

A. Gács and Z. Weiner, On $(q+t)$-arcs of type $(0, 2, t)$, Des. Codes Cryptogr., 29 (2003), 131-139.  doi: 10.1023/A:1024152424893.

[20]

G. Korchmáros and F. Mazzocca, On $(q+t)$-arcs of type $(0, 2, t)$ in a desarguesian plane of order $q$, Math. Proc. Camb. Philos. Soc., 108 (1990), 445-459.  doi: 10.1017/S0305004100069346.

[21]

I. N. Landjev, Linear codes over finite fields and finite projective geometries, Discrete Math., 213 (2000), 211-244.  doi: 10.1016/S0012-365X(99)00183-1.

[22]

M. LavrauwG. MarinoO. Polverino and R. Trombetti, Solution to an isotopism question concerning rank $2$ semifields, J. Combin. Des., 23 (2015), 60-77.  doi: 10.1002/jcd.21382.

[23]

M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, Contemp. Math, Amer. Math. Soc., Providence, RI, 632 (2015), 271–293. doi: 10.1090/conm/632/12633.

[24]

G. Longobardi, G. Marino, R. Trombetti and Y. Zhou, A large family of maximum scattered linear sets of ${\text{PG}}(1, q^n)$ and their associated MRD codes, arXiv: 2102.08287.

[25]

G. Longobardi and C. Zanella, Linear sets and MRD-codes arising from a class of scattered linearized polynomials, J. Algebr. Combin., 53 (2021), 639-661.  doi: 10.1007/s10801-020-01011-9.

[26]

G. Lunardon and O. Polverino, Blocking sets of size $q^t + q^{t-1} + 1$, J. Combin. Theory Ser. A, 90 (2000), 148-158.  doi: 10.1006/jcta.1999.3022.

[27]

G. MarinoM. Montanucci and F. Zullo, MRD-codes arising from the trinomial $x^q + x^{q^3}+ cx^{q^5} \in {\mathbb F}_{q^6}[x]$, Linear Algebra Appl., 591 (2020), 99-114.  doi: 10.1016/j.laa.2020.01.004.

[28]

V. Napolitano, O. Polverino, P. Santonastaso and F. Zullo, Linear sets on the projective line with complementary weights, arXiv: 2107.10641.

[29]

V. Napolitano, O. Polverino, P. Santonastaso and F. Zullo, Classifications and constructions of minimum size linear sets, submitted.

[30]

V. Napolitano and F. Zullo, Codes with few weights arising from linear sets, Adv. Math. Commun.. doi: 10.3934/amc.2020129.

[31]

A. Neri, P. Santonastaso and F. Zullo, Extending two families of maximum rank distance codes, arXiv: 2104.07602.

[32]

O. Polverino, Linear sets in finite projective spaces, Discrete Math., 310 (2010), 3096-3107.  doi: 10.1016/j.disc.2009.04.007.

[33]

O. Polverino and F. Zullo, On the number of roots of some linearized polynomials, Linear Algebra Appl., 601 (2020), 189-218.  doi: 10.1016/j.laa.2020.05.009.

[34]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.

[35]

M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Des. Codes Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7.

[36]

M. A. Tsfasman, S. G. Vlǎduţ and D. Nogin, Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, 139. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/139.

[37]

C. Zanella and F. Zullo, Vertex properties of maximum scattered linear sets of $ {\text{PG}}(1, q^n)$, Discrete Math., 343 (2020), 111800, 14 pp. doi: 10.1016/j.disc.2019.111800.

[38]

G. Zini and F. Zullo, Scattered subspaces and related codes, Des. Codes Cryptogr., 89 (2021), 1853-1873.  doi: 10.1007/s10623-021-00891-7.

Table 1.  Known examples of scattered $ \mathcal{D}_{f} $
$ n $ $ f(x) $ conditions ref.
i) $ x^{q^s} $ $ \gcd(s,n)=1 $ [6]
ii) $ x^{q^s}+\delta x^{q^{s(n-1)}} $ $ \begin{array}{cc} \gcd(s,n)=1,\\ \mathrm{N}_{q^n/q}(\delta)\neq 1 \end{array} $ [22,26]
iii) $ 2\ell $ $ \begin{array}{lll}x^{q^s}+x^{q^{s(\ell-1)}}+ \delta^{q^\ell+1}x^{q^{s(\ell+1)}}+ \delta^{1-q^{2\ell-1}}x^{q^{s(2\ell-1)}} \end{array} $ $ \begin{array}{cc} q \quad \text{odd},\\ \mathrm{N}_{q^{2\ell}/q^\ell}(\delta)=-1,\\ \gcd(s,t)=1 \end{array} $ [5,24,25,31,37]
iv) $ 6 $ $ x^q+\delta x^{q^{4}} $ $ \begin{array}{cc} q>4, \\ \text{certain choices of} \, \delta \end{array} $ [4,12,33]
v) $ 6 $ $ x^{q}+x^{q^3}+\delta x^{q^5} $ $ \begin{array}{cccc}q \quad \text{odd}, \\ \delta^2+\delta =1 \end{array} $ [13,27]
vi) $ 8 $ $ x^{q}+\delta x^{q^5} $ $ \begin{array}{cc} q\,\text{odd},\\ \delta^2=-1\end{array} $ [12]
$ n $ $ f(x) $ conditions ref.
i) $ x^{q^s} $ $ \gcd(s,n)=1 $ [6]
ii) $ x^{q^s}+\delta x^{q^{s(n-1)}} $ $ \begin{array}{cc} \gcd(s,n)=1,\\ \mathrm{N}_{q^n/q}(\delta)\neq 1 \end{array} $ [22,26]
iii) $ 2\ell $ $ \begin{array}{lll}x^{q^s}+x^{q^{s(\ell-1)}}+ \delta^{q^\ell+1}x^{q^{s(\ell+1)}}+ \delta^{1-q^{2\ell-1}}x^{q^{s(2\ell-1)}} \end{array} $ $ \begin{array}{cc} q \quad \text{odd},\\ \mathrm{N}_{q^{2\ell}/q^\ell}(\delta)=-1,\\ \gcd(s,t)=1 \end{array} $ [5,24,25,31,37]
iv) $ 6 $ $ x^q+\delta x^{q^{4}} $ $ \begin{array}{cc} q>4, \\ \text{certain choices of} \, \delta \end{array} $ [4,12,33]
v) $ 6 $ $ x^{q}+x^{q^3}+\delta x^{q^5} $ $ \begin{array}{cccc}q \quad \text{odd}, \\ \delta^2+\delta =1 \end{array} $ [13,27]
vi) $ 8 $ $ x^{q}+\delta x^{q^5} $ $ \begin{array}{cc} q\,\text{odd},\\ \delta^2=-1\end{array} $ [12]
Table 2.  Known examples of $ \mathcal{D}_{f} $ having exactly two points of weight $ n/2 $ and all the others of weight one
$ n $ $ f(x) $ conditions ref.
i) $ 2t $ $ \begin{array}{ll}\mathrm{Tr}_{q^n/q^t}\left( f\left(\frac{x}{\epsilon^{q^t}-\epsilon} \right) \right)+\\ \mathrm{Tr}_{q^n/q^t}\left( \frac{\epsilon^{q^t} x}{\epsilon^{q^t}-\epsilon}\right)\end{array} $ $ \begin{array}{cccc} \{1,\epsilon\}\,\, {\mathbb F}_{q^t}\text{-basis of } \mathbb{F}_{q^n}\\ f(z)=\sum_{i=0}^{t-1} A_iz^{q^i} \in \mathcal{L}_{t,q}\\ \text{ scattered}\end{array} $ [28]
ii) $ 2t $ $ \begin{array}{ll}\sum_{k=0}^{n-1}\left( \sum_{\ell=0}^{t-1} (u_\ell+\right.\\ \left. u_\ell^{q^s}\xi ){\lambda_{\ell}^*}^{q^k} \right)x^{q^k}\end{array} $ $ \begin{array}{ccc} \gcd(s,t)=1\\ \{1,\xi\}\,\, {\mathbb F}_{q^t}\text{-basis of } \mathbb{F}_{q^n}\\ \mu \in {\mathbb F}_{q^t} \colon \mathrm{N}_{q^t/q}(\mu)\neq 1\\ \mathrm{N}_{q^{t}/q}(-\xi^{q^t+1}\mu)\neq (-1)^t\\ \{u_0,\ldots,u_{t-1}\}\,\, {\mathbb F}_{q}\text{-basis of } {\mathbb F}_{q^t}\\ \mathcal{B}=(u_0+\mu u_0^{q^s}\xi,\ldots,\\ u_{t-1}+ \mu u_{t-1}^{q^s}\xi,u_0+u_0^{q^s}\xi,\\ \ldots,u_{t-1}+u_{t-1}^{q^s}\xi)\\ {\mathbb F}_{q}\text{-basis } \mathbb{F}_{q^n}\\ (\lambda_0^*,\ldots,\lambda_{n-1}^*)\text{ dual basis of } \mathbb{F}_{q^n} \end{array} $ [28]
$ n $ $ f(x) $ conditions ref.
i) $ 2t $ $ \begin{array}{ll}\mathrm{Tr}_{q^n/q^t}\left( f\left(\frac{x}{\epsilon^{q^t}-\epsilon} \right) \right)+\\ \mathrm{Tr}_{q^n/q^t}\left( \frac{\epsilon^{q^t} x}{\epsilon^{q^t}-\epsilon}\right)\end{array} $ $ \begin{array}{cccc} \{1,\epsilon\}\,\, {\mathbb F}_{q^t}\text{-basis of } \mathbb{F}_{q^n}\\ f(z)=\sum_{i=0}^{t-1} A_iz^{q^i} \in \mathcal{L}_{t,q}\\ \text{ scattered}\end{array} $ [28]
ii) $ 2t $ $ \begin{array}{ll}\sum_{k=0}^{n-1}\left( \sum_{\ell=0}^{t-1} (u_\ell+\right.\\ \left. u_\ell^{q^s}\xi ){\lambda_{\ell}^*}^{q^k} \right)x^{q^k}\end{array} $ $ \begin{array}{ccc} \gcd(s,t)=1\\ \{1,\xi\}\,\, {\mathbb F}_{q^t}\text{-basis of } \mathbb{F}_{q^n}\\ \mu \in {\mathbb F}_{q^t} \colon \mathrm{N}_{q^t/q}(\mu)\neq 1\\ \mathrm{N}_{q^{t}/q}(-\xi^{q^t+1}\mu)\neq (-1)^t\\ \{u_0,\ldots,u_{t-1}\}\,\, {\mathbb F}_{q}\text{-basis of } {\mathbb F}_{q^t}\\ \mathcal{B}=(u_0+\mu u_0^{q^s}\xi,\ldots,\\ u_{t-1}+ \mu u_{t-1}^{q^s}\xi,u_0+u_0^{q^s}\xi,\\ \ldots,u_{t-1}+u_{t-1}^{q^s}\xi)\\ {\mathbb F}_{q}\text{-basis } \mathbb{F}_{q^n}\\ (\lambda_0^*,\ldots,\lambda_{n-1}^*)\text{ dual basis of } \mathbb{F}_{q^n} \end{array} $ [28]
Table 3.  Known examples of $ \mathcal{D}_{f} $ which are $ i $-club with $ i>1 $
$ n $ $ i $ $ f(x) $ conditions ref.
i) $ rt $ $ r(t-1) $ $ \mathrm{Tr}_{q^{rt}/q^r}\circ x^{q^s} $ $ \gcd(s,n)=1 $ [15,20]
ii) $ n-2 $ $ \mathrm{Tr}_{q^n/q}(b_{n-2}x)+\lambda\mathrm{Tr}_{q^n/q}(b_{n-1}x) $ $ \begin{array}{ccc}\gcd(s,n)=1\\ {\mathbb F}_{q}(\lambda)= \mathbb{F}_{q^n}\\ \mathcal{B}=(1,\lambda,\ldots,\lambda^{n-3},\lambda^{n-2}+\omega,\omega\lambda)\\ \mathcal{B}\,\, {\mathbb F}_{q}\text{-basis of } \mathbb{F}_{q^n}\\ (b_0,\ldots,b_{n-1})\,\,\,\text{dual basis of } \mathcal{B} \end{array} $ [15,29]
iii) $ rt $ $ r(t-1) $ $ f\left( \mathrm{Tr}_{q^n/q^r}(c_0x) \right)-a\mathrm{Tr}_{q^n/q^r}(c_0x) $ $ \begin{array}{ccc} f(x) \in \mathcal{L}_{r,q}\,\, \text{scattered}\\ \{1,\omega,\ldots,\omega^{t-1}\}\,\,\, {\mathbb F}_{q^r}\text{-basis of } {\mathbb F}_{q^n}\\ c_0=\frac{1}{g'(\omega)}\sum_{j=0}^{t-1}\omega^j a_{i+j+1}\\ g(x)=a_0+a_1x+\ldots+a_{t-1}x^{t-1}+x^t\\ g(x)\text{ minimal polynomial of }\omega\text{ over } {\mathbb F}_{q^r}\\ f(x)-ax\,\, \text{is invertible over}\,\, {\mathbb F}_{q^r}\end{array} $ [15,19,29]
iv) $ rt $ $ r(t-1)+1 $ $ f\left( \mathrm{Tr}_{q^n/q^r}(c_0x) \right)-a\mathrm{Tr}_{q^n/q^r}(c_0x) $ $ \begin{array}{ccc} f(x) \in \mathcal{L}_{r,q}\,\, \text{scattered} \\ \{1,\omega,\ldots,\omega^{t-1}\}\,\,\, {\mathbb F}_{q^r}\text{-basis of } {\mathbb F}_{q^n}\\ c_0=\frac{1}{g'(\omega)}\sum_{j=0}^{t-1}\omega^j a_{i+j+1}\\ g(x)=a_0+a_1x+\ldots+a_{t-1}x^{t-1}+x^t\\ g(x)\text{ minimal polynomial of }\omega\text{ over } {\mathbb F}_{q^r}\\ f(x)-ax\,\, \text{is not invertible over}\,\, {\mathbb F}_{q^r} \end{array} $ [15,19,29]
$ n $ $ i $ $ f(x) $ conditions ref.
i) $ rt $ $ r(t-1) $ $ \mathrm{Tr}_{q^{rt}/q^r}\circ x^{q^s} $ $ \gcd(s,n)=1 $ [15,20]
ii) $ n-2 $ $ \mathrm{Tr}_{q^n/q}(b_{n-2}x)+\lambda\mathrm{Tr}_{q^n/q}(b_{n-1}x) $ $ \begin{array}{ccc}\gcd(s,n)=1\\ {\mathbb F}_{q}(\lambda)= \mathbb{F}_{q^n}\\ \mathcal{B}=(1,\lambda,\ldots,\lambda^{n-3},\lambda^{n-2}+\omega,\omega\lambda)\\ \mathcal{B}\,\, {\mathbb F}_{q}\text{-basis of } \mathbb{F}_{q^n}\\ (b_0,\ldots,b_{n-1})\,\,\,\text{dual basis of } \mathcal{B} \end{array} $ [15,29]
iii) $ rt $ $ r(t-1) $ $ f\left( \mathrm{Tr}_{q^n/q^r}(c_0x) \right)-a\mathrm{Tr}_{q^n/q^r}(c_0x) $ $ \begin{array}{ccc} f(x) \in \mathcal{L}_{r,q}\,\, \text{scattered}\\ \{1,\omega,\ldots,\omega^{t-1}\}\,\,\, {\mathbb F}_{q^r}\text{-basis of } {\mathbb F}_{q^n}\\ c_0=\frac{1}{g'(\omega)}\sum_{j=0}^{t-1}\omega^j a_{i+j+1}\\ g(x)=a_0+a_1x+\ldots+a_{t-1}x^{t-1}+x^t\\ g(x)\text{ minimal polynomial of }\omega\text{ over } {\mathbb F}_{q^r}\\ f(x)-ax\,\, \text{is invertible over}\,\, {\mathbb F}_{q^r}\end{array} $ [15,19,29]
iv) $ rt $ $ r(t-1)+1 $ $ f\left( \mathrm{Tr}_{q^n/q^r}(c_0x) \right)-a\mathrm{Tr}_{q^n/q^r}(c_0x) $ $ \begin{array}{ccc} f(x) \in \mathcal{L}_{r,q}\,\, \text{scattered} \\ \{1,\omega,\ldots,\omega^{t-1}\}\,\,\, {\mathbb F}_{q^r}\text{-basis of } {\mathbb F}_{q^n}\\ c_0=\frac{1}{g'(\omega)}\sum_{j=0}^{t-1}\omega^j a_{i+j+1}\\ g(x)=a_0+a_1x+\ldots+a_{t-1}x^{t-1}+x^t\\ g(x)\text{ minimal polynomial of }\omega\text{ over } {\mathbb F}_{q^r}\\ f(x)-ax\,\, \text{is not invertible over}\,\, {\mathbb F}_{q^r} \end{array} $ [15,19,29]
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