MinT - Best Known (56, 56+31, s)-Nets in Base 16

Best Known (56, 56+31, s)-Nets in Base 16

(56, 56+31, 583)-Net over F₁₆ — Constructive and digital

Digital (56, 87, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 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 (35, 66, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 33, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(56, 56+31, 594)-Net in Base 16 — Constructive

(56, 87, 594)-net in base 16, using

16¹ times duplication [i] based on (55, 86, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (9, 24, 80)-net in base 16, using
    - base change [i] based on digital (1, 16, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (31, 62, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 31, 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]

(56, 56+31, 2508)-Net over F₁₆ — Digital

Digital (56, 87, 2508)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+31, 3430011)-Net in Base 16 — Upper bound on s

There is no (56, 87, 3430012)-net in base 16, because

1 times m-reduction [i] would yield (56, 86, 3430012)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 835961 403808 236041 842118 159189 723947 688105 301127 546201 197268 517427 447032 697362 265694 114214 696234 491076 > 16⁸⁶ [i]