MinT - Best Known (39, 39+41, s)-Nets in Base 16

Best Known (39, 39+41, s)-Nets in Base 16

(39, 39+41, 130)-Net over F₁₆ — Constructive and digital

Digital (39, 80, 130)-net over F₁₆, using

13 times m-reduction [i] based on digital (39, 93, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 33, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (6, 60, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(39, 39+41, 192)-Net in Base 16 — Constructive

(39, 80, 192)-net in base 16, using

4 times m-reduction [i] based on (39, 84, 192)-net in base 16, using
- base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(39, 39+41, 266)-Net over F₁₆ — Digital

Digital (39, 80, 266)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+41, 31574)-Net in Base 16 — Upper bound on s

There is no (39, 80, 31575)-net in base 16, because

1 times m-reduction [i] would yield (39, 79, 31575)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133520 235763 285540 798485 906838 359578 008419 514217 490470 674294 770767 613861 101958 701737 069518 716251 > 16⁷⁹ [i]