MinT - Best Known (20, 45, s)-Nets in Base 16

Best Known (20, 45, s)-Nets in Base 16

(20, 45, 98)-Net over F₁₆ — Constructive and digital

Digital (20, 45, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 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, 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]

(20, 45, 129)-Net in Base 16 — Constructive

(20, 45, 129)-net in base 16, using

base change [i] based on (5, 30, 129)-net in base 64, using
- 5 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
  - base change [i] based on digital (0, 30, 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]

(20, 45, 129)-Net over F₁₆ — Digital

Digital (20, 45, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 45, 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]

(20, 45, 133)-Net in Base 16

(20, 45, 133)-net in base 16, using

base change [i] based on digital (5, 30, 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]

(20, 45, 9163)-Net in Base 16 — Upper bound on s

There is no (20, 45, 9164)-net in base 16, because

1 times m-reduction [i] would yield (20, 44, 9164)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 95798 786346 962610 918530 128502 952150 550082 109207 866896 > 16⁴⁴ [i]