MinT - Best Known (29−12, 29, s)-Nets in Base 16

Best Known (29−12, 29, s)-Nets in Base 16

Digital (17, 29, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (17, 30, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 15, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (17, 29, 645)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²⁹, 645, F₁₆, 12) (dual of [645, 616, 13]-code), using
- 2 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0) [i] based on linear OA(16²⁸, 642, F₁₆, 12) (dual of [642, 614, 13]-code), using
  - trace code [i] based on linear OA(256¹⁴, 321, F₂₅₆, 12) (dual of [321, 307, 13]-code), using
    - extended algebraic-geometric code AG_e(F,308P) [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 (17, 29, 131837)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 83079 953957 685992 918048 174917 007956 > 16²⁹ [i]