## Abstract

Let

polynomials of index t are those for which U is a maximum scattered linear set in PG(1, qmr) for infinitely many m. The classifications of exceptional scattered monic polynomials of index 0 (for q > 5) and of index 1 were obtained in [1]. In this paper we complete the classifications of exceptional scattered monic polynomials of index 0 for q ≤ 4. Also, some partial classifications are obtained for arbitrary t. As a consequence, the classification of exceptional scattered monic polynomials of index 2 is given.

*f*(X) ∈**F**_{q}^{r}[X] be a*q*-polynomial. If the Fq-subspace U = {(*x*^{qt},*f*(*x*)) | x ∈**F**_{q}^{n}} defines a maximum scattered linear set, then we call*f*(X) a scattered polynomial of index t. The asymptotic behavior of scattered polynomials of index t is an interesting open problem. In this sense, exceptional scatteredpolynomials of index t are those for which U is a maximum scattered linear set in PG(1, qmr) for infinitely many m. The classifications of exceptional scattered monic polynomials of index 0 (for q > 5) and of index 1 were obtained in [1]. In this paper we complete the classifications of exceptional scattered monic polynomials of index 0 for q ≤ 4. Also, some partial classifications are obtained for arbitrary t. As a consequence, the classification of exceptional scattered monic polynomials of index 2 is given.

Original language | English |
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Article number | 105386 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 179 |

Number of pages | 28 |

ISSN | 0097-3165 |

DOIs | |

Publication status | Published - 2021 |

## Keywords

- Maximum scattered linear set
- MRD code
- Algebraic curve
- Hasse-Weil bound