MinT - Best Known (82, 82+159, s)-Nets in Base 3

Best Known (82, 82+159, s)-Nets in Base 3

(82, 82+159, 57)-Net over F₃ — Constructive and digital

Digital (82, 241, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-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]

(82, 82+159, 84)-Net over F₃ — Digital

Digital (82, 241, 84)-net over F₃, using

t-expansion [i] based on digital (71, 241, 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]

(82, 82+159, 260)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 241, 261)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴¹, 261, F₃, 159) (dual of [261, 20, 160]-code), but
- residual code [i] would yield OA(3⁸², 101, S₃, 53), but
  - the linear programming bound shows that MÂ â‰¥ 10993 355000 871367 562211 653536 372020 125189 985096 / 6 532165 > 3⁸² [i]

(82, 82+159, 352)-Net in Base 3 — Upper bound on s

There is no (82, 241, 353)-net in base 3, because

1 times m-reduction [i] would yield (82, 240, 353)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 586670 155692 373394 634027 815743 005939 381806 094520 005571 603035 546372 868572 150594 973706 690953 209496 758420 451767 955179 > 3²⁴⁰ [i]