MinT - Best Known (78−54, 78, s)-Nets in Base 16

Best Known (78−54, 78, s)-Nets in Base 16

(78−54, 78, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 78, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 78, 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]

(78−54, 78, 98)-Net in Base 16 — Constructive

(24, 78, 98)-net in base 16, using

7 times m-reduction [i] based on (24, 85, 98)-net in base 16, using
- base change [i] based on digital (7, 68, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(78−54, 78, 129)-Net over F₁₆ — Digital

Digital (24, 78, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 78, 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]

(78−54, 78, 2177)-Net in Base 16 — Upper bound on s

There is no (24, 78, 2178)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8400 791942 859321 063961 808913 799818 284116 766120 030841 865103 129047 686801 940279 606909 477031 482816 > 16⁷⁸ [i]