MinT - Best Known (56, 56+86, s)-Nets in Base 9

Best Known (56, 56+86, s)-Nets in Base 9

(56, 56+86, 81)-Net over F₉ — Constructive and digital

Digital (56, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 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+86, 84)-Net in Base 9 — Constructive

(56, 142, 84)-net in base 9, using

2 times m-reduction [i] based on (56, 144, 84)-net in base 9, using
- base change [i] based on digital (8, 96, 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]

(56, 56+86, 182)-Net over F₉ — Digital

Digital (56, 142, 182)-net over F₉, using

t-expansion [i] based on digital (50, 142, 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+86, 2963)-Net in Base 9 — Upper bound on s

There is no (56, 142, 2964)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3217 372612 683983 878609 581535 195960 452715 103833 172774 882210 448127 422436 698696 607639 873322 990592 938763 226158 571522 714661 619862 212952 053345 > 9¹⁴² [i]