MinT - Best Known (63−37, 63, s)-Nets in Base 16

Best Known (63−37, 63, s)-Nets in Base 16

(63−37, 63, 98)-Net over F₁₆ — Constructive and digital

Digital (26, 63, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 43, 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]

(63−37, 63, 128)-Net in Base 16 — Constructive

(26, 63, 128)-net in base 16, using

base change [i] based on digital (5, 42, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(63−37, 63, 150)-Net over F₁₆ — Digital

Digital (26, 63, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(63−37, 63, 7062)-Net in Base 16 — Upper bound on s

There is no (26, 63, 7063)-net in base 16, because

1 times m-reduction [i] would yield (26, 62, 7063)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 452 722531 948726 984088 225796 045057 965828 262721 068375 306985 112934 883028 352811 > 16⁶² [i]