MinT - Best Known (63−33, 63, s)-Nets in Base 16

Best Known (63−33, 63, s)-Nets in Base 16

(63−33, 63, 130)-Net over F₁₆ — Constructive and digital

Digital (30, 63, 130)-net over F₁₆, using

3 times m-reduction [i] based on digital (30, 66, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 24, 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]
  2. digital (6, 42, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(63−33, 63, 192)-Net in Base 16 — Constructive

(30, 63, 192)-net in base 16, using

base change [i] based on digital (3, 36, 192)-net over F₁₂₈, using
- net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
    - a function field by SÃ©mirat [i]

(63−33, 63, 199)-Net over F₁₆ — Digital

Digital (30, 63, 199)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−33, 63, 209)-Net in Base 16

(30, 63, 209)-net in base 16, using

base change [i] based on digital (9, 42, 209)-net over F₆₄, using
- net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
    - a curve from the manYPoints database [i]

(63−33, 63, 21000)-Net in Base 16 — Upper bound on s

There is no (30, 63, 21001)-net in base 16, because

1 times m-reduction [i] would yield (30, 62, 21001)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 452 390804 520647 507981 965052 895722 485424 638546 943792 673498 827944 256495 091616 > 16⁶² [i]