MinT - Best Known (46, 81, s)-Nets in Base 16

Best Known (46, 81, s)-Nets in Base 16

Digital (46, 81, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (46, 82, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 41, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (46, 81, 699)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸¹, 699, F₁₆, 35) (dual of [699, 618, 36]-code), using
- 50 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 13 times 0, 1, 27 times 0) [i] based on linear OA(16⁷⁴, 642, F₁₆, 35) (dual of [642, 568, 36]-code), using
  - trace code [i] based on linear OA(256³⁷, 321, F₂₅₆, 35) (dual of [321, 284, 36]-code), using
    - extended algebraic-geometric code AG_e(F,285P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

There is no (46, 81, 221965)-net in base 16, because

1 times m-reduction [i] would yield (46, 80, 221965)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136107 765917 643962 294292 276527 732875 825154 088854 712153 209148 259584 764559 160374 874025 490687 920576 > 16⁸⁰ [i]