MinT - Best Known (78, 78+150, s)-Nets in Base 3

Best Known (78, 78+150, s)-Nets in Base 3

(78, 78+150, 53)-Net over F₃ — Constructive and digital

Digital (78, 228, 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+150, 84)-Net over F₃ — Digital

Digital (78, 228, 84)-net over F₃, using

t-expansion [i] based on digital (71, 228, 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+150, 250)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 228, 251)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁸, 251, F₃, 150) (dual of [251, 23, 151]-code), but
- residual code [i] would yield OA(3⁷⁸, 100, S₃, 50), but
  - the linear programming bound shows that MÂ â‰¥ 38101 440069 622014 833427 457285 684571 162543 165529 / 2181 771007 > 3⁷⁸ [i]

(78, 78+150, 336)-Net in Base 3 — Upper bound on s

There is no (78, 228, 337)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 045295 020935 262510 416069 842520 997215 254294 837737 322253 869969 442694 563274 733858 499794 564651 055587 309353 787835 > 3²²⁸ [i]