MinT - Best Known (57−29, 57, s)-Nets in Base 9

Best Known (57−29, 57, s)-Nets in Base 9

(57−29, 57, 82)-Net over F₉ — Constructive and digital

Digital (28, 57, 82)-net over F₉, using

base reduction for projective spaces (embedding PG(28,81) in PG(56,9)) for nets [i] based on digital (0, 29, 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]

(57−29, 57, 88)-Net in Base 9 — Constructive

(28, 57, 88)-net in base 9, using

base change [i] based on digital (9, 38, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(57−29, 57, 119)-Net over F₉ — Digital

Digital (28, 57, 119)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−29, 57, 4950)-Net in Base 9 — Upper bound on s

There is no (28, 57, 4951)-net in base 9, because

1 times m-reduction [i] would yield (28, 56, 4951)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 274580 900311 466003 490295 833234 689277 324134 487573 562641 > 9⁵⁶ [i]