MinT - Best Known (51, 51+33, s)-Nets in Base 16

Best Known (51, 51+33, s)-Nets in Base 16

(51, 51+33, 547)-Net over F₁₆ — Constructive and digital

Digital (51, 84, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 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 (33, 66, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 33, 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]

(51, 51+33, 1251)-Net over F₁₆ — Digital

Digital (51, 84, 1251)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+33, 799494)-Net in Base 16 — Upper bound on s

There is no (51, 84, 799495)-net in base 16, because

1 times m-reduction [i] would yield (51, 83, 799495)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8749 173708 081389 903441 344633 238763 709865 920469 140182 467639 121969 057443 039670 688225 386146 389818 908176 > 16⁸³ [i]