MinT - Best Known (57, 57+116, s)-Nets in Base 3

Best Known (57, 57+116, s)-Nets in Base 3

(57, 57+116, 48)-Net over F₃ — Constructive and digital

Digital (57, 173, 48)-net over F₃, using

t-expansion [i] based on digital (45, 173, 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+116, 64)-Net over F₃ — Digital

Digital (57, 173, 64)-net over F₃, using

t-expansion [i] based on digital (49, 173, 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+116, 182)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 173, 183)-net over F₃, because

2 times m-reduction [i] would yield digital (57, 171, 183)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷¹, 183, F₃, 114) (dual of [183, 12, 115]-code), but
  - residual code [i] would yield linear OA(3⁵⁷, 68, F₃, 38) (dual of [68, 11, 39]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(57, 57+116, 244)-Net in Base 3 — Upper bound on s

There is no (57, 173, 245)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41644 317570 530993 909684 840828 795442 749086 457814 623517 823213 138296 978171 569280 297489 > 3¹⁷³ [i]