MinT - Best Known (129−81, 129, s)-Nets in Base 3

Best Known (129−81, 129, s)-Nets in Base 3

(129−81, 129, 48)-Net over F₃ — Constructive and digital

Digital (48, 129, 48)-net over F₃, using

t-expansion [i] based on digital (45, 129, 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]

(129−81, 129, 56)-Net over F₃ — Digital

Digital (48, 129, 56)-net over F₃, using

t-expansion [i] based on digital (40, 129, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(129−81, 129, 226)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 129, 227)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁹, 227, F₃, 81) (dual of [227, 98, 82]-code), but
- residual code [i] would yield OA(3⁴⁸, 145, S₃, 27), but
  - the linear programming bound shows that MÂ â‰¥ 1072 673873 218950 092233 712509 383389 471446 292679 / 12873 944093 167948 027789 > 3⁴⁸ [i]

(129−81, 129, 227)-Net in Base 3 — Upper bound on s

There is no (48, 129, 228)-net in base 3, because

1 times m-reduction [i] would yield (48, 128, 228)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 865821 676882 308604 093985 394399 335188 247590 030570 742124 824257 > 3¹²⁸ [i]