MinT - Best Known (78−49, 78, s)-Nets in Base 16

Best Known (78−49, 78, s)-Nets in Base 16

(78−49, 78, 71)-Net over F₁₆ — Constructive and digital

Digital (29, 78, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 26, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (3, 52, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(78−49, 78, 120)-Net in Base 16 — Constructive

(29, 78, 120)-net in base 16, using

12 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
- base change [i] based on digital (11, 72, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(78−49, 78, 161)-Net over F₁₆ — Digital

Digital (29, 78, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(78−49, 78, 4756)-Net in Base 16 — Upper bound on s

There is no (29, 78, 4757)-net in base 16, because

1 times m-reduction [i] would yield (29, 77, 4757)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 522 545611 775599 085548 473254 817746 997989 999362 106184 628590 675212 958517 314972 256354 926104 539696 > 16⁷⁷ [i]