MinT - Best Known (75−39, 75, s)-Nets in Base 16

Best Known (75−39, 75, s)-Nets in Base 16

(75−39, 75, 130)-Net over F₁₆ — Constructive and digital

Digital (36, 75, 130)-net over F₁₆, using

9 times m-reduction [i] based on digital (36, 84, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 30, 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, 54, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(75−39, 75, 192)-Net in Base 16 — Constructive

(36, 75, 192)-net in base 16, using

2 times m-reduction [i] based on (36, 77, 192)-net in base 16, using
- base change [i] based on digital (3, 44, 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]

(75−39, 75, 235)-Net over F₁₆ — Digital

Digital (36, 75, 235)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−39, 75, 25863)-Net in Base 16 — Upper bound on s

There is no (36, 75, 25864)-net in base 16, because

1 times m-reduction [i] would yield (36, 74, 25864)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127388 995184 382820 332066 251660 766191 943758 697990 774531 669968 205629 991652 773402 813106 653616 > 16⁷⁴ [i]