MinT - Best Known (78, 78+159, s)-Nets in Base 3

Best Known (78, 78+159, s)-Nets in Base 3

(78, 78+159, 53)-Net over F₃ — Constructive and digital

Digital (78, 237, 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+159, 84)-Net over F₃ — Digital

Digital (78, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 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+159, 245)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 237, 246)-net over F₃, because

3 times m-reduction [i] would yield digital (78, 234, 246)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁴, 246, F₃, 156) (dual of [246, 12, 157]-code), but
  - residual code [i] would yield OA(3⁷⁸, 89, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 298 630446 198644 459587 118144 486403 305957 001183 / 16 135903 > 3⁷⁸ [i]

(78, 78+159, 329)-Net in Base 3 — Upper bound on s

There is no (78, 237, 330)-net in base 3, because

1 times m-reduction [i] would yield (78, 236, 330)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 42264 761970 691721 781577 502993 127790 182902 606103 820411 885622 758424 828061 984323 317788 063908 095345 477011 025088 839545 > 3²³⁶ [i]