MinT - Best Known (51, 51+110, s)-Nets in Base 3

Best Known (51, 51+110, s)-Nets in Base 3

(51, 51+110, 48)-Net over F₃ — Constructive and digital

Digital (51, 161, 48)-net over F₃, using

t-expansion [i] based on digital (45, 161, 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+110, 64)-Net over F₃ — Digital

Digital (51, 161, 64)-net over F₃, using

t-expansion [i] based on digital (49, 161, 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+110, 162)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 161, 163)-net over F₃, because

2 times m-reduction [i] would yield digital (51, 159, 163)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁹, 163, F₃, 108) (dual of [163, 4, 109]-code), but
  - Griesmer bound [i]

(51, 51+110, 215)-Net in Base 3 — Upper bound on s

There is no (51, 161, 216)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 70654 193378 741539 399703 167073 571047 367404 560459 081949 227532 295684 437433 795873 > 3¹⁶¹ [i]