MinT - Best Known (77−55, 77, s)-Nets in Base 16

Best Known (77−55, 77, s)-Nets in Base 16

(77−55, 77, 65)-Net over F₁₆ — Constructive and digital

Digital (22, 77, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 77, 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]

(77−55, 77, 76)-Net in Base 16 — Constructive

(22, 77, 76)-net in base 16, using

8 times m-reduction [i] based on (22, 85, 76)-net in base 16, using
- base change [i] based on digital (5, 68, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(77−55, 77, 129)-Net over F₁₆ — Digital

Digital (22, 77, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 77, 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]

(77−55, 77, 1770)-Net in Base 16 — Upper bound on s

There is no (22, 77, 1771)-net in base 16, because

1 times m-reduction [i] would yield (22, 76, 1771)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 867333 745338 729051 082611 053768 093547 946899 858532 789653 388953 849633 512475 620326 292212 969856 > 16⁷⁶ [i]