MinT - Best Known (32, 32+23, s)-Nets in Base 16

Best Known (32, 32+23, s)-Nets in Base 16

Digital (32, 55, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (32, 56, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 28, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (32, 55, 683)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁵, 683, F₁₆, 23) (dual of [683, 628, 24]-code), using
- 36 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 8 times 0, 1, 23 times 0) [i] based on linear OA(16⁵⁰, 642, F₁₆, 23) (dual of [642, 592, 24]-code), using
  - trace code [i] based on linear OA(256²⁵, 321, F₂₅₆, 23) (dual of [321, 296, 24]-code), using
    - extended algebraic-geometric code AG_e(F,297P) [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 (32, 55, 266716)-net in base 16, because

1 times m-reduction [i] would yield (32, 54, 266716)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 105315 085348 550434 859838 015479 457330 165768 866041 209107 798831 438641 > 16⁵⁴ [i]