MinT - Best Known (76−41, 76, s)-Nets in Base 16

Best Known (76−41, 76, s)-Nets in Base 16

(76−41, 76, 130)-Net over F₁₆ — Constructive and digital

Digital (35, 76, 130)-net over F₁₆, using

5 times m-reduction [i] based on digital (35, 81, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 29, 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 (6, 52, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(76−41, 76, 177)-Net in Base 16 — Constructive

(35, 76, 177)-net in base 16, using

8 times m-reduction [i] based on (35, 84, 177)-net in base 16, using
- base change [i] based on digital (7, 56, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(76−41, 76, 197)-Net over F₁₆ — Digital

Digital (35, 76, 197)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−41, 76, 209)-Net in Base 16

(35, 76, 209)-net in base 16, using

2 times m-reduction [i] based on (35, 78, 209)-net in base 16, using
- base change [i] based on digital (9, 52, 209)-net over F₆₄, using
  - net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
      - a curve from the manYPoints database [i]

(76−41, 76, 18130)-Net in Base 16 — Upper bound on s

There is no (35, 76, 18131)-net in base 16, because

1 times m-reduction [i] would yield (35, 75, 18131)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 038558 718850 698140 366346 689267 066499 532042 971855 426625 262597 463988 495510 612439 134589 400176 > 16⁷⁵ [i]