MinT - Best Known (26, 26+39, s)-Nets in Base 16

Best Known (26, 26+39, s)-Nets in Base 16

(26, 26+39, 89)-Net over F₁₆ — Constructive and digital

Digital (26, 65, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 45, 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]

(26, 26+39, 120)-Net in Base 16 — Constructive

(26, 65, 120)-net in base 16, using

10 times m-reduction [i] based on (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]

(26, 26+39, 150)-Net over F₁₆ — Digital

Digital (26, 65, 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]

(26, 26+39, 6002)-Net in Base 16 — Upper bound on s

There is no (26, 65, 6003)-net in base 16, because

1 times m-reduction [i] would yield (26, 64, 6003)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115806 227824 603183 421386 544250 747838 349106 419872 184996 504355 683286 573457 345456 > 16⁶⁴ [i]