MinT - Best Known (234−156, 234, s)-Nets in Base 3

Best Known (234−156, 234, s)-Nets in Base 3

(234−156, 234, 53)-Net over F₃ — Constructive and digital

Digital (78, 234, 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]

(234−156, 234, 84)-Net over F₃ — Digital

Digital (78, 234, 84)-net over F₃, using

t-expansion [i] based on digital (71, 234, 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]

(234−156, 234, 245)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 234, 246)-net over F₃, because

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]

(234−156, 234, 331)-Net in Base 3 — Upper bound on s

There is no (78, 234, 332)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5181 867711 566469 109180 306076 907434 466999 124354 630023 190096 369427 995199 615460 771987 828736 056273 625180 195990 457865 > 3²³⁴ [i]