MinT - Best Known (25, 78, s)-Nets in Base 16

Best Known (25, 78, s)-Nets in Base 16

(25, 78, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 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]

(25, 78, 104)-Net in Base 16 — Constructive

(25, 78, 104)-net in base 16, using

2 times m-reduction [i] based on (25, 80, 104)-net in base 16, using
- base change [i] based on digital (9, 64, 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]

(25, 78, 144)-Net over F₁₆ — Digital

Digital (25, 78, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(25, 78, 2575)-Net in Base 16 — Upper bound on s

There is no (25, 78, 2576)-net in base 16, because

1 times m-reduction [i] would yield (25, 77, 2576)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 524 177892 365666 959719 057205 259536 435426 765919 572490 601536 909953 580513 445295 092198 764615 634016 > 16⁷⁷ [i]