MinT - Best Known (81, 81+45, s)-Nets in Base 9

Best Known (81, 81+45, s)-Nets in Base 9

(81, 81+45, 740)-Net over F₉ — Constructive and digital

Digital (81, 126, 740)-net over F₉, using

4 times m-reduction [i] based on digital (81, 130, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 65, 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]

(81, 81+45, 1187)-Net over F₉ — Digital

Digital (81, 126, 1187)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+45, 298934)-Net in Base 9 — Upper bound on s

There is no (81, 126, 298935)-net in base 9, because

1 times m-reduction [i] would yield (81, 125, 298935)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 190688 918669 745831 668989 990768 441634 960887 243914 059522 923262 662160 647765 920357 637600 148274 155413 817459 337987 227031 086289 > 9¹²⁵ [i]