MinT - Best Known (56, 56+89, s)-Nets in Base 9

Best Known (56, 56+89, s)-Nets in Base 9

(56, 56+89, 81)-Net over F₉ — Constructive and digital

Digital (56, 145, 81)-net over F₉, using

t-expansion [i] based on digital (32, 145, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(56, 56+89, 82)-Net in Base 9 — Constructive

(56, 145, 82)-net in base 9, using

2 times m-reduction [i] based on (56, 147, 82)-net in base 9, using
- base change [i] based on digital (7, 98, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(56, 56+89, 182)-Net over F₉ — Digital

Digital (56, 145, 182)-net over F₉, using

t-expansion [i] based on digital (50, 145, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(56, 56+89, 2835)-Net in Base 9 — Upper bound on s

There is no (56, 145, 2836)-net in base 9, because

1 times m-reduction [i] would yield (56, 144, 2836)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 257633 991201 048644 559339 562155 169429 632909 534416 054183 529380 857390 194452 435838 295198 527563 440583 352840 975744 289624 760218 635303 628072 283265 > 9¹⁴⁴ [i]