MinT - Best Known (52, 52+39, s)-Nets in Base 16

Best Known (52, 52+39, s)-Nets in Base 16

Digital (52, 91, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (52, 92, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 46, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (52, 91, 786)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹¹, 786, F₁₆, 39) (dual of [786, 695, 40]-code), using
- 135 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 24 times 0, 1, 38 times 0, 1, 49 times 0) [i] based on linear OA(16⁸², 642, F₁₆, 39) (dual of [642, 560, 40]-code), using
  - trace code [i] based on linear OA(256⁴¹, 321, F₂₅₆, 39) (dual of [321, 280, 40]-code), using
    - extended algebraic-geometric code AG_e(F,281P) [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 (52, 91, 267199)-net in base 16, because

1 times m-reduction [i] would yield (52, 90, 267199)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 348648 828294 380926 886273 385627 947295 326843 644808 111047 372557 639540 260931 552027 292199 056394 968066 014217 941716 > 16⁹⁰ [i]