MinT - Best Known (189−126, 189, s)-Nets in Base 3

Best Known (189−126, 189, s)-Nets in Base 3

(189−126, 189, 48)-Net over F₃ — Constructive and digital

Digital (63, 189, 48)-net over F₃, using

t-expansion [i] based on digital (45, 189, 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]

(189−126, 189, 64)-Net over F₃ — Digital

Digital (63, 189, 64)-net over F₃, using

t-expansion [i] based on digital (49, 189, 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]

(189−126, 189, 200)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 189, 201)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁹, 201, F₃, 126) (dual of [201, 12, 127]-code), but
- residual code [i] would yield OA(3⁶³, 74, S₃, 42), but
  - the linear programming bound shows that MÂ â‰¥ 12 034337 282839 174538 591498 251772 066709 / 10 069525 > 3⁶³ [i]

(189−126, 189, 270)-Net in Base 3 — Upper bound on s

There is no (63, 189, 271)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 785911 325032 763864 393231 280108 387233 438204 972680 339569 946951 270468 987569 690027 693147 661763 > 3¹⁸⁹ [i]