MinT - Best Known (21, 21+59, s)-Nets in Base 16

Best Known (21, 21+59, s)-Nets in Base 16

(21, 21+59, 65)-Net over F₁₆ — Constructive and digital

Digital (21, 80, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 80, 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]

(21, 21+59, 76)-Net in Base 16 — Constructive

(21, 80, 76)-net in base 16, using

base change [i] based on digital (5, 64, 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]

(21, 21+59, 129)-Net over F₁₆ — Digital

Digital (21, 80, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 80, 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]

(21, 21+59, 1467)-Net in Base 16 — Upper bound on s

There is no (21, 80, 1468)-net in base 16, because

1 times m-reduction [i] would yield (21, 79, 1468)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 135132 812661 244862 332201 896616 549466 960285 499476 038601 969895 807367 702911 360653 379759 299515 910056 > 16⁷⁹ [i]