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Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

  • *Corresponding author: Ferdinando Zullo

    *Corresponding author: Ferdinando Zullo

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)

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  • 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.

    Mathematics Subject Classification: Primary: 511T71. Secondary: 11T06, 94B05.


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  • 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]
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    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]
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    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]
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