MinT - Best Known (87−37, 87, s)-Nets in Base 16

Best Known (87−37, 87, s)-Nets in Base 16

Digital (50, 87, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (50, 88, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 44, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (50, 87, 793)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁷, 793, F₁₆, 37) (dual of [793, 706, 38]-code), using
- 142 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 26 times 0, 1, 40 times 0, 1, 52 times 0) [i] based on linear OA(16⁷⁸, 642, F₁₆, 37) (dual of [642, 564, 38]-code), using
  - trace code [i] based on linear OA(256³⁹, 321, F₂₅₆, 37) (dual of [321, 282, 38]-code), using
    - extended algebraic-geometric code AG_e(F,283P) [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 (50, 87, 285130)-net in base 16, because

1 times m-reduction [i] would yield (50, 86, 285130)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 837691 820160 086073 865045 593452 312590 001540 006705 524351 657537 761131 346303 816880 014767 676899 300637 328976 > 16⁸⁶ [i]