MinT - Best Known (21, 21+27, s)-Nets in Base 16

Best Known (21, 21+27, s)-Nets in Base 16

(21, 21+27, 98)-Net over F₁₆ — Constructive and digital

Digital (21, 48, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 33, 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+27, 129)-Net in Base 16 — Constructive

(21, 48, 129)-net in base 16, using

1 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+27, 129)-Net over F₁₆ — Digital

Digital (21, 48, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 48, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(21, 21+27, 133)-Net in Base 16

(21, 48, 133)-net in base 16, using

base change [i] based on digital (5, 32, 133)-net over F₆₄, using
- net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
    - a curve from the manYPoints database [i]

(21, 21+27, 8517)-Net in Base 16 — Upper bound on s

There is no (21, 48, 8518)-net in base 16, because

1 times m-reduction [i] would yield (21, 47, 8518)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 392 503752 800944 558832 832987 797757 294887 959131 162825 718936 > 16⁴⁷ [i]