MinT - Best Known (66−42, 66, s)-Nets in Base 16

Best Known (66−42, 66, s)-Nets in Base 16

(66−42, 66, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 66, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 66, 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]

(66−42, 66, 104)-Net in Base 16 — Constructive

(24, 66, 104)-net in base 16, using

9 times m-reduction [i] based on (24, 75, 104)-net in base 16, using
- base change [i] based on digital (9, 60, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(66−42, 66, 129)-Net over F₁₆ — Digital

Digital (24, 66, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 66, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(66−42, 66, 3510)-Net in Base 16 — Upper bound on s

There is no (24, 66, 3511)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 727625 350982 706343 296796 831573 118479 828343 212721 336445 097117 307710 424273 156966 > 16⁶⁶ [i]