MinT - Best Known (57, 155, s)-Nets in Base 3

Best Known (57, 155, s)-Nets in Base 3

(57, 155, 48)-Net over F₃ — Constructive and digital

Digital (57, 155, 48)-net over F₃, using

t-expansion [i] based on digital (45, 155, 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, 155, 64)-Net over F₃ — Digital

Digital (57, 155, 64)-net over F₃, using

t-expansion [i] based on digital (49, 155, 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, 155, 257)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 155, 258)-net over F₃, because

2 times m-reduction [i] would yield digital (57, 153, 258)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵³, 258, F₃, 96) (dual of [258, 105, 97]-code), but
  - residual code [i] would yield OA(3⁵⁷, 161, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 721213 722132 726616 769951 419548 792269 427090 005182 815014 600979 / 455 415925 085809 684165 452127 356779 > 3⁵⁷ [i]

(57, 155, 263)-Net in Base 3 — Upper bound on s

There is no (57, 155, 264)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 307107 919339 301216 822055 943564 094425 424006 311396 616269 091111 122321 538321 > 3¹⁵⁵ [i]