MinT - Best Known (42−23, 42, s)-Nets in Base 16

Best Known (42−23, 42, s)-Nets in Base 16

(42−23, 42, 98)-Net over F₁₆ — Constructive and digital

Digital (19, 42, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 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, 29, 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]

(42−23, 42, 129)-Net over F₁₆ — Digital

Digital (19, 42, 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]

(42−23, 42, 150)-Net in Base 16 — Constructive

(19, 42, 150)-net in base 16, using

base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
- net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(42−23, 42, 10063)-Net in Base 16 — Upper bound on s

There is no (19, 42, 10064)-net in base 16, because

1 times m-reduction [i] would yield (19, 41, 10064)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 23 390816 327546 793565 496425 002279 720129 375837 681811 > 16⁴¹ [i]