MinT - Best Known (27, 27+28, s)-Nets in Base 9

Best Known (27, 27+28, s)-Nets in Base 9

(27, 27+28, 82)-Net over F₉ — Constructive and digital

Digital (27, 55, 82)-net over F₉, using

base reduction for projective spaces (embedding PG(27,81) in PG(54,9)) for nets [i] based on digital (0, 28, 82)-net over F₈₁, using
- net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
    - the rational function field F₈₁(x) [i]
  - Niederreiter sequence [i]

(27, 27+28, 84)-Net in Base 9 — Constructive

(27, 55, 84)-net in base 9, using

2 times m-reduction [i] based on (27, 57, 84)-net in base 9, using
- base change [i] based on digital (8, 38, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(27, 27+28, 116)-Net over F₉ — Digital

Digital (27, 55, 116)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+28, 4229)-Net in Base 9 — Upper bound on s

There is no (27, 55, 4230)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30440 822225 899865 865280 242383 447887 658959 719684 450401 > 9⁵⁵ [i]