MinT - Best Known (121−47, 121, s)-Nets in Base 16

Best Known (121−47, 121, s)-Nets in Base 16

(121−47, 121, 559)-Net over F₁₆ — Constructive and digital

Digital (74, 121, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 27, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (47, 94, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 47, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(121−47, 121, 1787)-Net over F₁₆ — Digital

Digital (74, 121, 1787)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(121−47, 121, 1204274)-Net in Base 16 — Upper bound on s

There is no (74, 121, 1204275)-net in base 16, because

1 times m-reduction [i] would yield (74, 120, 1204275)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 121755 700312 404425 896994 955131 229318 492351 442821 069208 295428 130513 354838 475217 648362 708202 728192 584950 164606 741147 358201 032368 494455 794462 297376 > 16¹²⁰ [i]