MinT - Best Known (76−53, 76, s)-Nets in Base 16

Best Known (76−53, 76, s)-Nets in Base 16

(76−53, 76, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 76, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 76, 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]

(76−53, 76, 98)-Net in Base 16 — Constructive

(23, 76, 98)-net in base 16, using

4 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]

(76−53, 76, 129)-Net over F₁₆ — Digital

Digital (23, 76, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 76, 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]

(76−53, 76, 2078)-Net in Base 16 — Upper bound on s

There is no (23, 76, 2079)-net in base 16, because

1 times m-reduction [i] would yield (23, 75, 2079)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 060063 541489 390996 026651 805560 676772 194685 183611 627835 850946 597270 478218 159797 451019 542061 > 16⁷⁵ [i]