MinT - Best Known (66, 197, s)-Nets in Base 3

Best Known (66, 197, s)-Nets in Base 3

(66, 197, 48)-Net over F₃ — Constructive and digital

Digital (66, 197, 48)-net over F₃, using

t-expansion [i] based on digital (45, 197, 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]

(66, 197, 64)-Net over F₃ — Digital

Digital (66, 197, 64)-net over F₃, using

t-expansion [i] based on digital (49, 197, 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]

(66, 197, 213)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 197, 214)-net over F₃, because

2 times m-reduction [i] would yield digital (66, 195, 214)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁵, 214, F₃, 129) (dual of [214, 19, 130]-code), but
  - residual code [i] would yield OA(3⁶⁶, 84, S₃, 43), but
    - the linear programming bound shows that MÂ â‰¥ 3447 047885 971598 137581 249186 171087 026769 / 88 665115 > 3⁶⁶ [i]

(66, 197, 283)-Net in Base 3 — Upper bound on s

There is no (66, 197, 284)-net in base 3, because

1 times m-reduction [i] would yield (66, 196, 284)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3314 731489 413341 812769 863625 147072 202337 635802 084232 266559 468953 802346 547107 774036 185653 922361 > 3¹⁹⁶ [i]