MinT - Best Known (57, 165, s)-Nets in Base 3

Best Known (57, 165, s)-Nets in Base 3

(57, 165, 48)-Net over F₃ — Constructive and digital

Digital (57, 165, 48)-net over F₃, using

t-expansion [i] based on digital (45, 165, 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, 165, 64)-Net over F₃ — Digital

Digital (57, 165, 64)-net over F₃, using

t-expansion [i] based on digital (49, 165, 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, 165, 200)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 165, 201)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁵, 201, F₃, 108) (dual of [201, 36, 109]-code), but
- residual code [i] would yield OA(3⁵⁷, 92, S₃, 36), but
  - the linear programming bound shows that MÂ â‰¥ 120 721890 049904 553965 455860 898229 550534 869060 685567 / 70191 477994 212742 316032 > 3⁵⁷ [i]

(57, 165, 250)-Net in Base 3 — Upper bound on s

There is no (57, 165, 251)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 315578 887658 422406 039689 345029 035600 443723 768072 020407 435211 429571 099421 420269 > 3¹⁶⁵ [i]