MinT - Best Known (54, 54+105, s)-Nets in Base 3

Best Known (54, 54+105, s)-Nets in Base 3

(54, 54+105, 48)-Net over F₃ — Constructive and digital

Digital (54, 159, 48)-net over F₃, using

t-expansion [i] based on digital (45, 159, 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+105, 64)-Net over F₃ — Digital

Digital (54, 159, 64)-net over F₃, using

t-expansion [i] based on digital (49, 159, 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+105, 182)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 159, 183)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁹, 183, F₃, 105) (dual of [183, 24, 106]-code), but
- residual code [i] would yield OA(3⁵⁴, 77, S₃, 35), but
  - the linear programming bound shows that MÂ â‰¥ 49772 879413 668128 111108 466099 084271 883739 / 760 672693 173302 > 3⁵⁴ [i]

(54, 54+105, 236)-Net in Base 3 — Upper bound on s

There is no (54, 159, 237)-net in base 3, because

1 times m-reduction [i] would yield (54, 158, 237)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2488 162787 225524 997990 215223 941656 684279 003637 311281 346054 364579 861081 994961 > 3¹⁵⁸ [i]