MinT - Best Known (52, 52+91, s)-Nets in Base 3

Best Known (52, 52+91, s)-Nets in Base 3

(52, 52+91, 48)-Net over F₃ — Constructive and digital

Digital (52, 143, 48)-net over F₃, using

t-expansion [i] based on digital (45, 143, 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]

(52, 52+91, 64)-Net over F₃ — Digital

Digital (52, 143, 64)-net over F₃, using

t-expansion [i] based on digital (49, 143, 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]

(52, 52+91, 227)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 143, 228)-net over F₃, because

1 times m-reduction [i] would yield digital (52, 142, 228)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴², 228, F₃, 90) (dual of [228, 86, 91]-code), but
  - residual code [i] would yield OA(3⁵², 137, S₃, 30), but
    - the linear programming bound shows that MÂ â‰¥ 44562 633012 596463 108701 497133 833862 064231 636556 224801 562500 / 6631 930927 956088 032510 798182 283163 > 3⁵² [i]

(52, 52+91, 240)-Net in Base 3 — Upper bound on s

There is no (52, 143, 241)-net in base 3, because

1 times m-reduction [i] would yield (52, 142, 241)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 61 042087 124024 157651 985472 118403 576766 951013 621009 628496 399806 147123 > 3¹⁴² [i]