MinT - Best Known (57, 57+112, s)-Nets in Base 3

Best Known (57, 57+112, s)-Nets in Base 3

(57, 57+112, 48)-Net over F₃ — Constructive and digital

Digital (57, 169, 48)-net over F₃, using

t-expansion [i] based on digital (45, 169, 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]

(57, 57+112, 64)-Net over F₃ — Digital

Digital (57, 169, 64)-net over F₃, using

t-expansion [i] based on digital (49, 169, 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]

(57, 57+112, 189)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 169, 190)-net over F₃, because

1 times m-reduction [i] would yield digital (57, 168, 190)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁸, 190, F₃, 111) (dual of [190, 22, 112]-code), but
  - residual code [i] would yield OA(3⁵⁷, 78, S₃, 37), but
    - the linear programming bound shows that MÂ â‰¥ 4 189427 328087 433604 993587 700284 480257 / 2119 390625 > 3⁵⁷ [i]

(57, 57+112, 247)-Net in Base 3 — Upper bound on s

There is no (57, 169, 248)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 492 389096 818886 926355 646322 378771 328013 861151 601312 349964 144635 418980 659533 389057 > 3¹⁶⁹ [i]