MinT - Best Known (226−148, 226, s)-Nets in Base 3

Best Known (226−148, 226, s)-Nets in Base 3

(226−148, 226, 53)-Net over F₃ — Constructive and digital

Digital (78, 226, 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]

(226−148, 226, 84)-Net over F₃ — Digital

Digital (78, 226, 84)-net over F₃, using

t-expansion [i] based on digital (71, 226, 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]

(226−148, 226, 259)-Net over F₃ — Upper bound on s (digital)

There is no digital (78, 226, 260)-net over F₃, because

1 times m-reduction [i] would yield digital (78, 225, 260)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁵, 260, F₃, 147) (dual of [260, 35, 148]-code), but
  - residual code [i] would yield OA(3⁷⁸, 112, S₃, 49), but
    - the linear programming bound shows that MÂ â‰¥ 23121 339982 129752 217216 969972 698951 685311 859808 619791 997981 / 1328 024731 338747 547105 > 3⁷⁸ [i]

(226−148, 226, 338)-Net in Base 3 — Upper bound on s

There is no (78, 226, 339)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 800752 634022 440866 201895 606905 382696 799069 377059 842115 962412 774517 591515 332912 119576 677730 774378 463492 928629 > 3²²⁶ [i]