MinT - Best Known (47−31, 47, s)-Nets in Base 16

Best Known (47−31, 47, s)-Nets in Base 16

(47−31, 47, 65)-Net over F₁₆ — Constructive and digital

Digital (16, 47, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 47, 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]

(47−31, 47, 76)-Net in Base 16 — Constructive

(16, 47, 76)-net in base 16, using

8 times m-reduction [i] based on (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]

(47−31, 47, 98)-Net over F₁₆ — Digital

Digital (16, 47, 98)-net over F₁₆, using

t-expansion [i] based on digital (15, 47, 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]

(47−31, 47, 2102)-Net in Base 16 — Upper bound on s

There is no (16, 47, 2103)-net in base 16, because

1 times m-reduction [i] would yield (16, 46, 2103)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 24 669948 372491 044338 188665 217511 059801 711409 484290 596176 > 16⁴⁶ [i]