MinT - Best Known (50, 50+96, s)-Nets in Base 3

Best Known (50, 50+96, s)-Nets in Base 3

(50, 50+96, 48)-Net over F₃ — Constructive and digital

Digital (50, 146, 48)-net over F₃, using

t-expansion [i] based on digital (45, 146, 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]

(50, 50+96, 64)-Net over F₃ — Digital

Digital (50, 146, 64)-net over F₃, using

t-expansion [i] based on digital (49, 146, 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]

(50, 50+96, 176)-Net over F₃ — Upper bound on s (digital)

There is no digital (50, 146, 177)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁶, 177, F₃, 96) (dual of [177, 31, 97]-code), but
- residual code [i] would yield OA(3⁵⁰, 80, S₃, 32), but
  - the linear programming bound shows that MÂ â‰¥ 17966 137866 413855 600211 208294 867018 069814 838765 / 21749 437213 575033 309776 > 3⁵⁰ [i]

(50, 50+96, 220)-Net in Base 3 — Upper bound on s

There is no (50, 146, 221)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4921 047673 729013 989684 142528 903786 324336 195642 047536 983221 207734 585153 > 3¹⁴⁶ [i]