MinT - Best Known (21, 21+25, s)-Nets in Base 16

Best Known (21, 21+25, s)-Nets in Base 16

(21, 21+25, 103)-Net over F₁₆ — Constructive and digital

Digital (21, 46, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 31, 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]

(21, 21+25, 129)-Net in Base 16 — Constructive

(21, 46, 129)-net in base 16, using

3 times m-reduction [i] based on (21, 49, 129)-net in base 16, using
- base change [i] based on digital (0, 28, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(21, 21+25, 134)-Net over F₁₆ — Digital

Digital (21, 46, 134)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(21, 21+25, 11547)-Net in Base 16 — Upper bound on s

There is no (21, 46, 11548)-net in base 16, because

1 times m-reduction [i] would yield (21, 45, 11548)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 533491 606359 173307 273793 844718 694148 933401 817155 864116 > 16⁴⁵ [i]