MinT - Best Known (65, 65+89, s)-Nets in Base 8

Best Known (65, 65+89, s)-Nets in Base 8

(65, 65+89, 99)-Net over F₈ — Constructive and digital

Digital (65, 154, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 51, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (14, 103, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(65, 65+89, 144)-Net over F₈ — Digital

Digital (65, 154, 144)-net over F₈, using

t-expansion [i] based on digital (45, 154, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(65, 65+89, 150)-Net in Base 8

(65, 154, 150)-net in base 8, using

2 times m-reduction [i] based on (65, 156, 150)-net in base 8, using
- base change [i] based on digital (26, 117, 150)-net over F₁₆, using
  - net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
      - a curve from the manYPoints database [i]

(65, 65+89, 3377)-Net in Base 8 — Upper bound on s

There is no (65, 154, 3378)-net in base 8, because

1 times m-reduction [i] would yield (65, 153, 3378)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 502763 126292 057484 441738 652056 826210 940490 825936 138768 786838 557848 680319 813883 669918 401687 000971 479782 949982 256482 771769 255569 649324 847204 > 8¹⁵³ [i]