MinT - Best Known (51, 51+104, s)-Nets in Base 3

Best Known (51, 51+104, s)-Nets in Base 3

(51, 51+104, 48)-Net over F₃ — Constructive and digital

Digital (51, 155, 48)-net over F₃, using

t-expansion [i] based on digital (45, 155, 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+104, 64)-Net over F₃ — Digital

Digital (51, 155, 64)-net over F₃, using

t-expansion [i] based on digital (49, 155, 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+104, 167)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 155, 168)-net over F₃, because

2 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+104, 219)-Net in Base 3 — Upper bound on s

There is no (51, 155, 220)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 97 566970 545879 617895 288724 978711 603928 315342 372511 872711 863861 734743 578497 > 3¹⁵⁵ [i]