MinT - Best Known (36−15, 36, s)-Nets in Base 16

Best Known (36−15, 36, s)-Nets in Base 16

Digital (21, 36, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 18, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

Digital (21, 36, 652)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁶, 652, F₁₆, 15) (dual of [652, 616, 16]-code), using
- 8 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 7 times 0) [i] based on linear OA(16³⁴, 642, F₁₆, 15) (dual of [642, 608, 16]-code), using
  - trace code [i] based on linear OA(256¹⁷, 321, F₂₅₆, 15) (dual of [321, 304, 16]-code), using
    - extended algebraic-geometric code AG_e(F,305P) [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 (21, 36, 236277)-net in base 16, because

1 times m-reduction [i] would yield (21, 35, 236277)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 393808 203623 308045 244239 179708 748145 405136 > 16³⁵ [i]