MinT - Best Known (54, 156, s)-Nets in Base 3

Best Known (54, 156, s)-Nets in Base 3

(54, 156, 48)-Net over F₃ — Constructive and digital

Digital (54, 156, 48)-net over F₃, using

t-expansion [i] based on digital (45, 156, 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, 156, 64)-Net over F₃ — Digital

Digital (54, 156, 64)-net over F₃, using

t-expansion [i] based on digital (49, 156, 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, 156, 193)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 156, 194)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁶, 194, F₃, 102) (dual of [194, 38, 103]-code), but
- residual code [i] would yield OA(3⁵⁴, 91, S₃, 34), but
  - the linear programming bound shows that MÂ â‰¥ 474 977782 539634 784051 796341 060967 928894 025210 202817 107183 513784 170892 978866 150753 / 7 995136 378245 997436 849186 466159 148790 740510 898632 999345 > 3⁵⁴ [i]

(54, 156, 238)-Net in Base 3 — Upper bound on s

There is no (54, 156, 239)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 278 420477 880798 026422 004823 387841 349895 182663 771015 021451 685601 885338 777179 > 3¹⁵⁶ [i]