MinT - Best Known (57, 156, s)-Nets in Base 3

Best Known (57, 156, s)-Nets in Base 3

(57, 156, 48)-Net over F₃ — Constructive and digital

Digital (57, 156, 48)-net over F₃, using

t-expansion [i] based on digital (45, 156, 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, 156, 64)-Net over F₃ — Digital

Digital (57, 156, 64)-net over F₃, using

t-expansion [i] based on digital (49, 156, 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, 156, 247)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 156, 248)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁶, 248, F₃, 99) (dual of [248, 92, 100]-code), but
- residual code [i] would yield OA(3⁵⁷, 148, S₃, 33), but
  - the linear programming bound shows that MÂ â‰¥ 339310 353108 625838 622205 253005 107344 580119 357971 883548 907069 767680 / 208 213577 356617 676998 634448 123348 499991 > 3⁵⁷ [i]

(57, 156, 263)-Net in Base 3 — Upper bound on s

There is no (57, 156, 264)-net in base 3, because

1 times m-reduction [i] would yield (57, 155, 264)-net in base 3, but
- 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]