MinT - Best Known (64, 64+126, s)-Nets in Base 3

Best Known (64, 64+126, s)-Nets in Base 3

(64, 64+126, 48)-Net over F₃ — Constructive and digital

Digital (64, 190, 48)-net over F₃, using

t-expansion [i] based on digital (45, 190, 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]

(64, 64+126, 64)-Net over F₃ — Digital

Digital (64, 190, 64)-net over F₃, using

t-expansion [i] based on digital (49, 190, 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]

(64, 64+126, 205)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 190, 206)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁰, 206, F₃, 126) (dual of [206, 16, 127]-code), but
- residual code [i] would yield OA(3⁶⁴, 79, S₃, 42), but
  - the linear programming bound shows that MÂ â‰¥ 1127 088319 766791 916099 558651 458916 971683 / 304 050635 > 3⁶⁴ [i]

(64, 64+126, 275)-Net in Base 3 — Upper bound on s

There is no (64, 190, 276)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 673270 650828 187289 607583 851950 215137 493459 790633 439125 001432 086303 179335 243450 592621 152113 > 3¹⁹⁰ [i]