MinT - Best Known (27, 27+51, s)-Nets in Base 16

Best Known (27, 27+51, s)-Nets in Base 16

(27, 27+51, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 78, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 78, 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]

(27, 27+51, 120)-Net in Base 16 — Constructive

(27, 78, 120)-net in base 16, using

2 times m-reduction [i] based on (27, 80, 120)-net in base 16, using
- base change [i] based on digital (11, 64, 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]

(27, 27+51, 156)-Net over F₁₆ — Digital

Digital (27, 78, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(27, 27+51, 3455)-Net in Base 16 — Upper bound on s

There is no (27, 78, 3456)-net in base 16, because

1 times m-reduction [i] would yield (27, 77, 3456)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 523 343701 490905 311266 081646 294059 033833 278553 103280 584468 263069 163375 848563 756254 686294 101001 > 16⁷⁷ [i]