MinT - Best Known (18, 18+43, s)-Nets in Base 16

Best Known (18, 18+43, s)-Nets in Base 16

(18, 18+43, 65)-Net over F₁₆ — Constructive and digital

Digital (18, 61, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 61, 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]

(18, 18+43, 76)-Net in Base 16 — Constructive

(18, 61, 76)-net in base 16, using

4 times m-reduction [i] based on (18, 65, 76)-net in base 16, using
- base change [i] based on digital (5, 52, 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]

(18, 18+43, 113)-Net over F₁₆ — Digital

Digital (18, 61, 113)-net over F₁₆, using

net from sequence [i] based on digital (18, 112)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 18 and N(F) ≥ 113, using
  - a curve from the manYPoints database [i]

(18, 18+43, 1583)-Net in Base 16 — Upper bound on s

There is no (18, 61, 1584)-net in base 16, because

1 times m-reduction [i] would yield (18, 60, 1584)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 777112 444961 901455 388463 368637 969194 113031 154563 909736 877294 030578 259586 > 16⁶⁰ [i]