MinT - Best Known (51, 51+103, s)-Nets in Base 3

Best Known (51, 51+103, s)-Nets in Base 3

(51, 51+103, 48)-Net over F₃ — Constructive and digital

Digital (51, 154, 48)-net over F₃, using

t-expansion [i] based on digital (45, 154, 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]

(51, 51+103, 64)-Net over F₃ — Digital

Digital (51, 154, 64)-net over F₃, using

t-expansion [i] based on digital (49, 154, 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]

(51, 51+103, 167)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 154, 168)-net over F₃, because

1 times m-reduction [i] would yield digital (51, 153, 168)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵³, 168, F₃, 102) (dual of [168, 15, 103]-code), but
  - residual code [i] would yield OA(3⁵¹, 65, S₃, 34), but
    - the linear programming bound shows that MÂ â‰¥ 342 054256 122719 546156 841781 498869 / 115 101637 > 3⁵¹ [i]

(51, 51+103, 221)-Net in Base 3 — Upper bound on s

There is no (51, 154, 222)-net in base 3, because

1 times m-reduction [i] would yield (51, 153, 222)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 792193 787208 875390 030382 417002 796433 950228 389847 567570 244975 113876 380065 > 3¹⁵³ [i]