MinT - Best Known (30−19, 30, s)-Nets in Base 16

Best Known (30−19, 30, s)-Nets in Base 16

(30−19, 30, 65)-Net over F₁₆ — Constructive and digital

Digital (11, 30, 65)-net over F₁₆, using

t-expansion [i] based on 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]

(30−19, 30, 80)-Net in Base 16 — Constructive

(11, 30, 80)-net in base 16, using

base change [i] based on digital (1, 20, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(30−19, 30, 81)-Net over F₁₆ — Digital

Digital (11, 30, 81)-net over F₁₆, using

t-expansion [i] based on digital (10, 30, 81)-net over F₁₆, using
- net from sequence [i] based on digital (10, 80)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 10 and N(F) ≥ 81, using
    - a curve from the manYPoints database [i]

(30−19, 30, 83)-Net in Base 16

(11, 30, 83)-net in base 16, using

base change [i] based on digital (5, 24, 83)-net over F₃₂, using
- net from sequence [i] based on digital (5, 82)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 83, using
    - a curve from the manYPoints database [i]

(30−19, 30, 2092)-Net in Base 16 — Upper bound on s

There is no (11, 30, 2093)-net in base 16, because

1 times m-reduction [i] would yield (11, 29, 2093)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 83283 148512 401640 097604 645117 365156 > 16²⁹ [i]