MinT - Best Known (125−78, 125, s)-Nets in Base 3

Best Known (125−78, 125, s)-Nets in Base 3

(125−78, 125, 48)-Net over F₃ — Constructive and digital

Digital (47, 125, 48)-net over F₃, using

t-expansion [i] based on digital (45, 125, 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]

(125−78, 125, 56)-Net over F₃ — Digital

Digital (47, 125, 56)-net over F₃, using

t-expansion [i] based on digital (40, 125, 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]

(125−78, 125, 223)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 125, 224)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹²⁵, 224, F₃, 78) (dual of [224, 99, 79]-code), but
- residual code [i] would yield OA(3⁴⁷, 145, S₃, 26), but
  - the linear programming bound shows that MÂ â‰¥ 831 528639 252434 333719 554172 193113 411271 544000 / 29642 853144 349355 775349 > 3⁴⁷ [i]

(125−78, 125, 224)-Net in Base 3 — Upper bound on s

There is no (47, 125, 225)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 500564 961536 755564 786324 654823 705902 211247 168021 100673 166219 > 3¹²⁵ [i]