MinT - Best Known (64−41, 64, s)-Nets in Base 16

Best Known (64−41, 64, s)-Nets in Base 16

(64−41, 64, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 64, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 64, 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]

(64−41, 64, 104)-Net in Base 16 — Constructive

(23, 64, 104)-net in base 16, using

6 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]

(64−41, 64, 129)-Net over F₁₆ — Digital

Digital (23, 64, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 64, 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]

(64−41, 64, 3426)-Net in Base 16 — Upper bound on s

There is no (23, 64, 3427)-net in base 16, because

1 times m-reduction [i] would yield (23, 63, 3427)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7267 328636 259557 516814 595352 346248 690347 299227 501812 055543 422198 984021 911976 > 16⁶³ [i]