MinT - Best Known (111−41, 111, s)-Nets in Base 16

Best Known (111−41, 111, s)-Nets in Base 16

(111−41, 111, 581)-Net over F₁₆ — Constructive and digital

Digital (70, 111, 581)-net over F₁₆, using

16¹ times duplication [i] based on digital (69, 110, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 26, 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, 84, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(111−41, 111, 2327)-Net over F₁₆ — Digital

Digital (70, 111, 2327)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−41, 111, 2322056)-Net in Base 16 — Upper bound on s

There is no (70, 111, 2322057)-net in base 16, because

1 times m-reduction [i] would yield (70, 110, 2322057)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 839217 166755 683227 138140 531105 758737 171812 799595 508439 004852 324567 264625 098006 123687 839793 994244 417210 337486 282916 290884 565503 000976 > 16¹¹⁰ [i]