MinT - Best Known (21, 21+37, s)-Nets in Base 16

Best Known (21, 21+37, s)-Nets in Base 16

(21, 21+37, 65)-Net over F₁₆ — Constructive and digital

Digital (21, 58, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 58, 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+37, 104)-Net in Base 16 — Constructive

(21, 58, 104)-net in base 16, using

2 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+37, 129)-Net over F₁₆ — Digital

Digital (21, 58, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 58, 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+37, 3264)-Net in Base 16 — Upper bound on s

There is no (21, 58, 3265)-net in base 16, because

1 times m-reduction [i] would yield (21, 57, 3265)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 432 883738 565075 344442 191836 759572 360871 895088 297392 555545 497187 398176 > 16⁵⁷ [i]