MinT - Best Known (56, 56+166, s)-Nets in Base 3

Best Known (56, 56+166, s)-Nets in Base 3

(56, 56+166, 48)-Net over F₃ — Constructive and digital

Digital (56, 222, 48)-net over F₃, using

t-expansion [i] based on digital (45, 222, 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]

(56, 56+166, 64)-Net over F₃ — Digital

Digital (56, 222, 64)-net over F₃, using

t-expansion [i] based on digital (49, 222, 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]

(56, 56+166, 177)-Net over F₃ — Upper bound on s (digital)

There is no digital (56, 222, 178)-net over F₃, because

49 times m-reduction [i] would yield digital (56, 173, 178)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷³, 178, F₃, 117) (dual of [178, 5, 118]-code), but
  - Griesmer bound [i]

(56, 56+166, 181)-Net in Base 3 — Upper bound on s

There is no (56, 222, 182)-net in base 3, because

45 times m-reduction [i] would yield (56, 177, 182)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷⁷, 182, S₃, 121), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 228 532044 137599 177017 869183 161846 685251 274404 207185 590172 004697 234871 412029 099114 058803 / 61 > 3¹⁷⁷ [i]