MinT - Best Known (53, 53+78, s)-Nets in Base 9

Best Known (53, 53+78, s)-Nets in Base 9

(53, 53+78, 81)-Net over F₉ — Constructive and digital

Digital (53, 131, 81)-net over F₉, using

t-expansion [i] based on digital (32, 131, 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]

(53, 53+78, 88)-Net in Base 9 — Constructive

(53, 131, 88)-net in base 9, using

1 times m-reduction [i] based on (53, 132, 88)-net in base 9, using
- base change [i] based on digital (9, 88, 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]

(53, 53+78, 182)-Net over F₉ — Digital

Digital (53, 131, 182)-net over F₉, using

t-expansion [i] based on digital (50, 131, 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]

(53, 53+78, 3063)-Net in Base 9 — Upper bound on s

There is no (53, 131, 3064)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101708 316497 998637 441464 138578 582669 674046 566765 024712 766401 571789 566259 243846 227271 594640 178353 025707 402276 678428 426986 072385 > 9¹³¹ [i]