MinT - Best Known (58, 58+116, s)-Nets in Base 3

Best Known (58, 58+116, s)-Nets in Base 3

(58, 58+116, 48)-Net over F₃ — Constructive and digital

Digital (58, 174, 48)-net over F₃, using

t-expansion [i] based on digital (45, 174, 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]

(58, 58+116, 64)-Net over F₃ — Digital

Digital (58, 174, 64)-net over F₃, using

t-expansion [i] based on digital (49, 174, 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]

(58, 58+116, 189)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 174, 190)-net over F₃, because

2 times m-reduction [i] would yield digital (58, 172, 190)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷², 190, F₃, 114) (dual of [190, 18, 115]-code), but
  - residual code [i] would yield OA(3⁵⁸, 75, S₃, 38), but
    - the linear programming bound shows that MÂ â‰¥ 425880 948792 153754 309607 908493 218857 / 85 592416 > 3⁵⁸ [i]

(58, 58+116, 249)-Net in Base 3 — Upper bound on s

There is no (58, 174, 250)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 110546 142022 811316 756552 953792 311865 513316 348288 618817 367916 505233 557599 707756 306261 > 3¹⁷⁴ [i]