MinT - Best Known (197−132, 197, s)-Nets in Base 3

Best Known (197−132, 197, s)-Nets in Base 3

(197−132, 197, 48)-Net over F₃ — Constructive and digital

Digital (65, 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]

(197−132, 197, 64)-Net over F₃ — Digital

Digital (65, 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]

(197−132, 197, 205)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 197, 206)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁷, 206, F₃, 132) (dual of [206, 9, 133]-code), but
- residual code [i] would yield linear OA(3⁶⁵, 73, F₃, 44) (dual of [73, 8, 45]-code), but
  - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(197−132, 197, 276)-Net in Base 3 — Upper bound on s

There is no (65, 197, 277)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10366 691643 761583 623655 385380 173443 492588 926518 843306 840351 001736 458516 894996 848767 510998 255345 > 3¹⁹⁷ [i]