MinT - Best Known (44, 75, s)-Nets in Base 16

Best Known (44, 75, s)-Nets in Base 16

Digital (44, 75, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 76, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 38, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (44, 75, 838)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁵, 838, F₁₆, 31) (dual of [838, 763, 32]-code), using
- 187 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 6 times 0, 1, 16 times 0, 1, 34 times 0, 1, 54 times 0, 1, 69 times 0) [i] based on linear OA(16⁶⁶, 642, F₁₆, 31) (dual of [642, 576, 32]-code), using
  - trace code [i] based on linear OA(256³³, 321, F₂₅₆, 31) (dual of [321, 288, 32]-code), using
    - extended algebraic-geometric code AG_e(F,289P) [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 (44, 75, 373243)-net in base 16, because

1 times m-reduction [i] would yield (44, 74, 373243)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127316 289503 285804 059669 444749 432886 370823 868501 614031 997085 354443 119085 994251 445839 060176 > 16⁷⁴ [i]