MinT - Best Known (70, 113, s)-Nets in Base 16

Best Known (70, 113, s)-Nets in Base 16

(70, 113, 579)-Net over F₁₆ — Constructive and digital

Digital (70, 113, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 27, 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]
2. digital (43, 86, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 43, 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]

(70, 113, 1932)-Net over F₁₆ — Digital

Digital (70, 113, 1932)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 113, 1528872)-Net in Base 16 — Upper bound on s

There is no (70, 113, 1528873)-net in base 16, because

1 times m-reduction [i] would yield (70, 112, 1528873)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 726 842251 104585 360308 892133 456687 994000 798189 624308 793432 673097 828805 900038 570000 768022 405182 386279 617569 883644 264132 299991 204200 048496 > 16¹¹² [i]