MinT - Best Known (21, 21+31, s)-Nets in Base 16

Best Known (21, 21+31, s)-Nets in Base 16

(21, 21+31, 82)-Net over F₁₆ — Constructive and digital

Digital (21, 52, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 37, 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+31, 104)-Net in Base 16 — Constructive

(21, 52, 104)-net in base 16, using

8 times m-reduction [i] based on (21, 60, 104)-net in base 16, using
- base change [i] based on digital (9, 48, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(21, 21+31, 129)-Net over F₁₆ — Digital

Digital (21, 52, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 52, 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+31, 5309)-Net in Base 16 — Upper bound on s

There is no (21, 52, 5310)-net in base 16, because

1 times m-reduction [i] would yield (21, 51, 5310)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 768574 920715 073430 121570 594696 827262 792270 083730 271744 150376 > 16⁵¹ [i]