MinT - Best Known (78−55, 78, s)-Nets in Base 16

Best Known (78−55, 78, s)-Nets in Base 16

(78−55, 78, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 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]

(78−55, 78, 98)-Net in Base 16 — Constructive

(23, 78, 98)-net in base 16, using

2 times m-reduction [i] based on (23, 80, 98)-net in base 16, using
- base change [i] based on digital (7, 64, 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]

(78−55, 78, 129)-Net over F₁₆ — Digital

Digital (23, 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]

(78−55, 78, 1963)-Net in Base 16 — Upper bound on s

There is no (23, 78, 1964)-net in base 16, because

1 times m-reduction [i] would yield (23, 77, 1964)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 525 009686 996076 182225 030123 017211 239361 530317 099924 142117 129668 834924 060607 106238 905929 059046 > 16⁷⁷ [i]