MinT - Best Known (86−31, 86, s)-Nets in Base 9

Best Known (86−31, 86, s)-Nets in Base 9

(86−31, 86, 344)-Net over F₉ — Constructive and digital

Digital (55, 86, 344)-net over F₉, using

10 times m-reduction [i] based on digital (55, 96, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 48, 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]

(86−31, 86, 834)-Net over F₉ — Digital

Digital (55, 86, 834)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−31, 86, 205131)-Net in Base 9 — Upper bound on s

There is no (55, 86, 205132)-net in base 9, because

1 times m-reduction [i] would yield (55, 85, 205132)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1290 129316 540955 378040 512530 234679 615646 631298 567507 632975 818056 024826 844006 890273 > 9⁸⁵ [i]