MinT - Best Known (231−152, 231, s)-Nets in Base 3

Best Known (231−152, 231, s)-Nets in Base 3

(231−152, 231, 54)-Net over F₃ — Constructive and digital

Digital (79, 231, 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]

(231−152, 231, 84)-Net over F₃ — Digital

Digital (79, 231, 84)-net over F₃, using

t-expansion [i] based on digital (71, 231, 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]

(231−152, 231, 258)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 231, 259)-net over F₃, because

2 times m-reduction [i] would yield digital (79, 229, 259)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁹, 259, F₃, 150) (dual of [259, 30, 151]-code), but
  - residual code [i] would yield OA(3⁷⁹, 108, S₃, 50), but
    - the linear programming bound shows that MÂ â‰¥ 17 534511 337853 558668 492502 167227 612507 359380 469822 536899 / 298348 310384 556250 > 3⁷⁹ [i]

(231−152, 231, 340)-Net in Base 3 — Upper bound on s

There is no (79, 231, 341)-net in base 3, because

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]