MinT - Best Known (51−21, 51, s)-Nets in Base 16

Best Known (51−21, 51, s)-Nets in Base 16

Digital (30, 51, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (30, 52, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 26, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (30, 51, 699)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵¹, 699, F₁₆, 21) (dual of [699, 648, 22]-code), using
- 52 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 12 times 0, 1, 34 times 0) [i] based on linear OA(16⁴⁶, 642, F₁₆, 21) (dual of [642, 596, 22]-code), using
  - trace code [i] based on linear OA(256²³, 321, F₂₅₆, 21) (dual of [321, 298, 22]-code), using
    - extended algebraic-geometric code AG_e(F,299P) [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 (30, 51, 316576)-net in base 16, because

1 times m-reduction [i] would yield (30, 50, 316576)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 606942 398660 718185 953702 757813 870125 226682 706118 404214 510901 > 16⁵⁰ [i]