MinT - Best Known (55−39, 55, s)-Nets in Base 16

Best Known (55−39, 55, s)-Nets in Base 16

(55−39, 55, 65)-Net over F₁₆ — Constructive and digital

Digital (16, 55, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 55, 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]

(55−39, 55, 76)-Net in Base 16 — Constructive

(16, 55, 76)-net in base 16, using

base change [i] based on digital (5, 44, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(55−39, 55, 98)-Net over F₁₆ — Digital

Digital (16, 55, 98)-net over F₁₆, using

t-expansion [i] based on digital (15, 55, 98)-net over F₁₆, using
- net from sequence [i] based on digital (15, 97)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 15 and N(F) ≥ 98, using
    - a curve from the manYPoints database [i]

(55−39, 55, 1387)-Net in Base 16 — Upper bound on s

There is no (16, 55, 1388)-net in base 16, because

1 times m-reduction [i] would yield (16, 54, 1388)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 106349 387535 966199 259913 148397 398669 867533 726492 775327 594387 714306 > 16⁵⁴ [i]