MinT - Best Known (231−171, 231, s)-Nets in Base 3

Best Known (231−171, 231, s)-Nets in Base 3

(231−171, 231, 48)-Net over F₃ — Constructive and digital

Digital (60, 231, 48)-net over F₃, using

t-expansion [i] based on digital (45, 231, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(231−171, 231, 64)-Net over F₃ — Digital

Digital (60, 231, 64)-net over F₃, using

t-expansion [i] based on digital (49, 231, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(231−171, 231, 190)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 231, 191)-net over F₃, because

48 times m-reduction [i] would yield digital (60, 183, 191)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸³, 191, F₃, 123) (dual of [191, 8, 124]-code), but
  - residual code [i] would yield OA(3⁶⁰, 67, S₃, 41), but
    - the linear programming bound shows that MÂ â‰¥ 367 404168 771298 835858 389852 553067 / 8575 > 3⁶⁰ [i]

(231−171, 231, 193)-Net in Base 3 — Upper bound on s

There is no (60, 231, 194)-net in base 3, because

42 times m-reduction [i] would yield (60, 189, 194)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁸⁹, 194, S₃, 129), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 121 451298 068529 844233 553416 568714 964256 622520 646270 917226 600348 303197 298080 156462 274524 325123 / 65 > 3¹⁸⁹ [i]