MinT - Best Known (123−45, 123, s)-Nets in Base 9

Best Known (123−45, 123, s)-Nets in Base 9

(123−45, 123, 740)-Net over F₉ — Constructive and digital

Digital (78, 123, 740)-net over F₉, using

1 times m-reduction [i] based on digital (78, 124, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 62, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(123−45, 123, 1025)-Net over F₉ — Digital

Digital (78, 123, 1025)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−45, 123, 221536)-Net in Base 9 — Upper bound on s

There is no (78, 123, 221537)-net in base 9, because

1 times m-reduction [i] would yield (78, 122, 221537)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 261 578767 189319 715511 691790 729319 985226 258253 468543 143184 215058 201496 151214 564724 512941 572513 085881 439255 024271 163505 > 9¹²² [i]