MinT - Best Known (53, 157, s)-Nets in Base 3

Best Known (53, 157, s)-Nets in Base 3

(53, 157, 48)-Net over F₃ — Constructive and digital

Digital (53, 157, 48)-net over F₃, using

t-expansion [i] based on digital (45, 157, 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]

(53, 157, 64)-Net over F₃ — Digital

Digital (53, 157, 64)-net over F₃, using

t-expansion [i] based on digital (49, 157, 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]

(53, 157, 184)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 157, 185)-net over F₃, because

2 times m-reduction [i] would yield digital (53, 155, 185)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁵, 185, F₃, 102) (dual of [185, 30, 103]-code), but
  - residual code [i] would yield OA(3⁵³, 82, S₃, 34), but
    - the linear programming bound shows that MÂ â‰¥ 109 763126 725714 200014 609915 641908 532996 574841 / 4 714638 489586 850000 > 3⁵³ [i]

(53, 157, 230)-Net in Base 3 — Upper bound on s

There is no (53, 157, 231)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 812 084991 583128 879698 351514 093005 597128 341281 954236 680572 474776 595152 834329 > 3¹⁵⁷ [i]