MinT - Best Known (69−44, 69, s)-Nets in Base 16

Best Known (69−44, 69, s)-Nets in Base 16

(69−44, 69, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 69, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 69, 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]

(69−44, 69, 120)-Net in Base 16 — Constructive

(25, 69, 120)-net in base 16, using

1 times m-reduction [i] based on (25, 70, 120)-net in base 16, using
- base change [i] based on digital (11, 56, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(69−44, 69, 144)-Net over F₁₆ — Digital

Digital (25, 69, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(69−44, 69, 3596)-Net in Base 16 — Upper bound on s

There is no (25, 69, 3597)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 121797 248340 993026 923608 856716 985654 445380 870338 498696 599107 334433 625702 371262 807936 > 16⁶⁹ [i]