MinT - Best Known (75−49, 75, s)-Nets in Base 16

Best Known (75−49, 75, s)-Nets in Base 16

(75−49, 75, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 75, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 75, 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]

(75−49, 75, 120)-Net in Base 16 — Constructive

(26, 75, 120)-net in base 16, using

base change [i] based on digital (11, 60, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(75−49, 75, 150)-Net over F₁₆ — Digital

Digital (26, 75, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(75−49, 75, 3359)-Net in Base 16 — Upper bound on s

There is no (26, 75, 3360)-net in base 16, because

1 times m-reduction [i] would yield (26, 74, 3360)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127686 758547 698776 922366 203066 503570 670854 042057 092606 818066 013085 817227 673022 407783 831351 > 16⁷⁴ [i]