MinT - Best Known (232−153, 232, s)-Nets in Base 3

Best Known (232−153, 232, s)-Nets in Base 3

(232−153, 232, 54)-Net over F₃ — Constructive and digital

Digital (79, 232, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-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]

(232−153, 232, 84)-Net over F₃ — Digital

Digital (79, 232, 84)-net over F₃, using

t-expansion [i] based on digital (71, 232, 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]

(232−153, 232, 251)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 232, 252)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³², 252, F₃, 153) (dual of [252, 20, 154]-code), but
- residual code [i] would yield OA(3⁷⁹, 98, S₃, 51), but
  - the linear programming bound shows that MÂ â‰¥ 37359 886424 244488 451760 980773 909770 698911 495751 / 677 763515 > 3⁷⁹ [i]

(232−153, 232, 340)-Net in Base 3 — Upper bound on s

There is no (79, 232, 341)-net in base 3, because

1 times m-reduction [i] would yield (79, 231, 341)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 188 196406 263815 738515 691849 352782 376449 148337 497740 489437 535910 855086 966296 392707 771983 595939 739064 996570 699185 > 3²³¹ [i]