MinT - Best Known (57−24, 57, s)-Nets in Base 16

Best Known (57−24, 57, s)-Nets in Base 16

Digital (33, 57, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (33, 58, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 29, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (33, 57, 677)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁷, 677, F₁₆, 24) (dual of [677, 620, 25]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 6 times 0, 1, 19 times 0) [i] based on linear OA(16⁵², 642, F₁₆, 24) (dual of [642, 590, 25]-code), using
  - trace code [i] based on linear OA(256²⁶, 321, F₂₅₆, 24) (dual of [321, 295, 25]-code), using
    - extended algebraic-geometric code AG_e(F,296P) [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 (33, 57, 184853)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 431 373828 708903 308954 716534 954604 501171 036133 598769 480301 391583 213266 > 16⁵⁷ [i]