MinT - Best Known (106−39, 106, s)-Nets in Base 16

Best Known (106−39, 106, s)-Nets in Base 16

(106−39, 106, 581)-Net over F₁₆ — Constructive and digital

Digital (67, 106, 581)-net over F₁₆, using

16¹ times duplication [i] based on digital (66, 105, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 25, 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 (41, 80, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 40, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(106−39, 106, 2308)-Net over F₁₆ — Digital

Digital (67, 106, 2308)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−39, 106, 2384897)-Net in Base 16 — Upper bound on s

There is no (67, 106, 2384898)-net in base 16, because

1 times m-reduction [i] would yield (67, 105, 2384898)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 707704 089792 923581 012628 861507 627485 337719 418042 256304 761598 990793 142369 621425 815837 507853 344484 404317 586060 814601 485399 430656 > 16¹⁰⁵ [i]