MinT - Best Known (25, 52, s)-Nets in Base 16

Best Known (25, 52, s)-Nets in Base 16

(25, 52, 130)-Net over F₁₆ — Constructive and digital

Digital (25, 52, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 19, 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, 33, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(25, 52, 177)-Net in Base 16 — Constructive

(25, 52, 177)-net in base 16, using

2 times m-reduction [i] based on (25, 54, 177)-net in base 16, using
- base change [i] based on digital (7, 36, 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]

(25, 52, 184)-Net over F₁₆ — Digital

Digital (25, 52, 184)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 52, 19999)-Net in Base 16 — Upper bound on s

There is no (25, 52, 20000)-net in base 16, because

1 times m-reduction [i] would yield (25, 51, 20000)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 717804 249156 744059 466533 950324 208269 291785 159165 817681 587501 > 16⁵¹ [i]