MinT - Best Known (85, 85+61, s)-Nets in Base 9

Best Known (85, 85+61, s)-Nets in Base 9

(85, 85+61, 344)-Net over F₉ — Constructive and digital

Digital (85, 146, 344)-net over F₉, using

t-expansion [i] based on digital (82, 146, 344)-net over F₉, using
- 4 times m-reduction [i] based on digital (82, 150, 344)-net over F₉, using
  - trace code for nets [i] based on digital (7, 75, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(85, 85+61, 602)-Net over F₉ — Digital

Digital (85, 146, 602)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+61, 61622)-Net in Base 9 — Upper bound on s

There is no (85, 146, 61623)-net in base 9, because

1 times m-reduction [i] would yield (85, 145, 61623)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 318367 647421 416532 284104 639540 912468 410266 083171 844624 769935 764058 534675 127455 464346 243060 830630 615330 820924 023679 335656 870520 945203 457681 > 9¹⁴⁵ [i]