MinT - Best Known (78, 78+113, s)-Nets in Base 3

Best Known (78, 78+113, s)-Nets in Base 3

(78, 78+113, 53)-Net over F₃ — Constructive and digital

Digital (78, 191, 53)-net over F₃, using

net from sequence [i] based on digital (78, 52)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 52)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(78, 78+113, 84)-Net over F₃ — Digital

Digital (78, 191, 84)-net over F₃, using

t-expansion [i] based on digital (71, 191, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(78, 78+113, 398)-Net in Base 3 — Upper bound on s

There is no (78, 191, 399)-net in base 3, because

1 times m-reduction [i] would yield (78, 190, 399)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 839131 867970 644861 752417 988573 170764 882231 812247 319142 585756 345528 932885 309093 247575 328401 > 3¹⁹⁰ [i]