MinT - Best Known (52, 52+99, s)-Nets in Base 3

Best Known (52, 52+99, s)-Nets in Base 3

(52, 52+99, 48)-Net over F₃ — Constructive and digital

Digital (52, 151, 48)-net over F₃, using

t-expansion [i] based on digital (45, 151, 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+99, 64)-Net over F₃ — Digital

Digital (52, 151, 64)-net over F₃, using

t-expansion [i] based on digital (49, 151, 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+99, 184)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 151, 185)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵¹, 185, F₃, 99) (dual of [185, 34, 100]-code), but
- residual code [i] would yield OA(3⁵², 85, S₃, 33), but
  - the linear programming bound shows that MÂ â‰¥ 860342 754412 435094 906698 858579 674571 922984 984079 / 127652 777684 493779 532100 > 3⁵² [i]

(52, 52+99, 230)-Net in Base 3 — Upper bound on s

There is no (52, 151, 231)-net in base 3, because

1 times m-reduction [i] would yield (52, 150, 231)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 389558 704943 699837 710035 014045 662347 290046 333616 494478 997634 119119 762287 > 3¹⁵⁰ [i]