MinT - Best Known (81−29, 81, s)-Nets in Base 16

Best Known (81−29, 81, s)-Nets in Base 16

(81−29, 81, 581)-Net over F₁₆ — Constructive and digital

Digital (52, 81, 581)-net over F₁₆, using

16¹ times duplication [i] based on digital (51, 80, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 20, 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 (31, 60, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(81−29, 81, 594)-Net in Base 16 — Constructive

(52, 81, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (9, 23, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (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 (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(81−29, 81, 2306)-Net over F₁₆ — Digital

Digital (52, 81, 2306)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−29, 81, 3062323)-Net in Base 16 — Upper bound on s

There is no (52, 81, 3062324)-net in base 16, because

1 times m-reduction [i] would yield (52, 80, 3062324)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 135993 690588 603205 710145 274595 767831 368951 617658 109425 519686 329427 044601 049558 546270 503573 124916 > 16⁸⁰ [i]