MinT - Best Known (125−49, 125, s)-Nets in Base 16

Best Known (125−49, 125, s)-Nets in Base 16

(125−49, 125, 552)-Net over F₁₆ — Constructive and digital

Digital (76, 125, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (49, 98, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 49, 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]

(125−49, 125, 1732)-Net over F₁₆ — Digital

Digital (76, 125, 1732)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−49, 125, 1087809)-Net in Base 16 — Upper bound on s

There is no (76, 125, 1087810)-net in base 16, because

1 times m-reduction [i] would yield (76, 124, 1087810)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 204589 429453 832415 615127 598885 199516 385756 937746 654812 961308 407855 743973 431365 543207 055836 616941 846606 320405 054094 559566 392107 230001 213001 459450 583976 > 16¹²⁴ [i]