MinT - Best Known (69−46, 69, s)-Nets in Base 16

Best Known (69−46, 69, s)-Nets in Base 16

(69−46, 69, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 69, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 69, 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]

(69−46, 69, 104)-Net in Base 16 — Constructive

(23, 69, 104)-net in base 16, using

1 times m-reduction [i] based on (23, 70, 104)-net in base 16, using
- base change [i] based on digital (9, 56, 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]

(69−46, 69, 129)-Net over F₁₆ — Digital

Digital (23, 69, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 69, 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]

(69−46, 69, 2562)-Net in Base 16 — Upper bound on s

There is no (23, 69, 2563)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 122317 731441 622555 454603 973904 361061 387363 859840 566395 121734 687796 911896 599352 577536 > 16⁶⁹ [i]