MinT - Best Known (67, 67+132, s)-Nets in Base 3

Best Known (67, 67+132, s)-Nets in Base 3

(67, 67+132, 48)-Net over F₃ — Constructive and digital

Digital (67, 199, 48)-net over F₃, using

t-expansion [i] based on digital (45, 199, 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]

(67, 67+132, 72)-Net over F₃ — Digital

Digital (67, 199, 72)-net over F₃, using

net from sequence [i] based on digital (67, 71)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(67, 67+132, 214)-Net over F₃ — Upper bound on s (digital)

There is no digital (67, 199, 215)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁹, 215, F₃, 132) (dual of [215, 16, 133]-code), but
- residual code [i] would yield OA(3⁶⁷, 82, S₃, 44), but
  - the linear programming bound shows that MÂ â‰¥ 11063 764601 718263 371177 932141 744468 313023 / 98 314060 > 3⁶⁷ [i]

(67, 67+132, 288)-Net in Base 3 — Upper bound on s

There is no (67, 199, 289)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 106892 150552 955265 459113 631322 737267 435544 331977 266018 927267 175464 381459 144830 937218 313719 597129 > 3¹⁹⁹ [i]