MinT - Best Known (40, 40+29, s)-Nets in Base 16

Best Known (40, 40+29, s)-Nets in Base 16

Digital (40, 69, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 70, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 35, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (40, 69, 724)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁹, 724, F₁₆, 29) (dual of [724, 655, 30]-code), using
- 75 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 8 times 0, 1, 20 times 0, 1, 40 times 0) [i] based on linear OA(16⁶², 642, F₁₆, 29) (dual of [642, 580, 30]-code), using
  - trace code [i] based on linear OA(256³¹, 321, F₂₅₆, 29) (dual of [321, 290, 30]-code), using
    - extended algebraic-geometric code AG_e(F,291P) [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 (40, 69, 284406)-net in base 16, because

1 times m-reduction [i] would yield (40, 68, 284406)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7588 758700 699847 348546 187605 663095 055391 676701 400795 907757 375832 547358 946676 133736 > 16⁶⁸ [i]