MinT - Best Known (185−122, 185, s)-Nets in Base 3

Best Known (185−122, 185, s)-Nets in Base 3

(185−122, 185, 48)-Net over F₃ — Constructive and digital

Digital (63, 185, 48)-net over F₃, using

t-expansion [i] based on digital (45, 185, 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]

(185−122, 185, 64)-Net over F₃ — Digital

Digital (63, 185, 64)-net over F₃, using

t-expansion [i] based on digital (49, 185, 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]

(185−122, 185, 213)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 185, 214)-net over F₃, because

2 times m-reduction [i] would yield digital (63, 183, 214)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸³, 214, F₃, 120) (dual of [214, 31, 121]-code), but
  - residual code [i] would yield OA(3⁶³, 93, S₃, 40), but
    - the linear programming bound shows that MÂ â‰¥ 3 468104 662873 063250 302984 700066 235894 540957 / 2 937118 328933 > 3⁶³ [i]

(185−122, 185, 273)-Net in Base 3 — Upper bound on s

There is no (63, 185, 274)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 20436 524157 726527 138275 743747 433931 685184 602066 959129 536675 907915 980417 458559 835917 614861 > 3¹⁸⁵ [i]