MinT - Best Known (54, 54+92, s)-Nets in Base 3

Best Known (54, 54+92, s)-Nets in Base 3

(54, 54+92, 48)-Net over F₃ — Constructive and digital

Digital (54, 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]

(54, 54+92, 64)-Net over F₃ — Digital

Digital (54, 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]

(54, 54+92, 249)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 146, 250)-net over F₃, because

2 times m-reduction [i] would yield digital (54, 144, 250)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁴, 250, F₃, 90) (dual of [250, 106, 91]-code), but
  - residual code [i] would yield OA(3⁵⁴, 159, S₃, 30), but
    - the linear programming bound shows that MÂ â‰¥ 57 512545 780273 008066 954941 653603 554048 685010 751297 100155 / 986015 082551 936588 824240 203061 > 3⁵⁴ [i]

(54, 54+92, 251)-Net in Base 3 — Upper bound on s

There is no (54, 146, 252)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5067 611600 482304 544719 122250 641382 706735 945083 038928 836179 567329 714985 > 3¹⁴⁶ [i]