MinT - Best Known (30, 30+77, s)-Nets in Base 16

Best Known (30, 30+77, s)-Nets in Base 16

(30, 30+77, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 107, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 107, 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]

(30, 30+77, 98)-Net in Base 16 — Constructive

(30, 107, 98)-net in base 16, using

8 times m-reduction [i] based on (30, 115, 98)-net in base 16, using
- base change [i] based on digital (7, 92, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(30, 30+77, 162)-Net over F₁₆ — Digital

Digital (30, 107, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(30, 30+77, 2267)-Net in Base 16 — Upper bound on s

There is no (30, 107, 2268)-net in base 16, because

1 times m-reduction [i] would yield (30, 106, 2268)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 589502 174743 331100 669250 924589 146474 946408 048649 368678 138115 123327 125007 632555 032881 912781 403788 698213 305953 439618 674437 569436 > 16¹⁰⁶ [i]