MinT - Best Known (174−115, 174, s)-Nets in Base 3

Best Known (174−115, 174, s)-Nets in Base 3

(174−115, 174, 48)-Net over F₃ — Constructive and digital

Digital (59, 174, 48)-net over F₃, using

t-expansion [i] based on digital (45, 174, 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]

(174−115, 174, 64)-Net over F₃ — Digital

Digital (59, 174, 64)-net over F₃, using

t-expansion [i] based on digital (49, 174, 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]

(174−115, 174, 197)-Net over F₃ — Upper bound on s (digital)

There is no digital (59, 174, 198)-net over F₃, because

1 times m-reduction [i] would yield digital (59, 173, 198)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷³, 198, F₃, 114) (dual of [198, 25, 115]-code), but
  - residual code [i] would yield OA(3⁵⁹, 83, S₃, 38), but
    - the linear programming bound shows that MÂ â‰¥ 325 626781 031292 288709 976539 068217 532079 726399 / 18937 606422 109375 > 3⁵⁹ [i]

(174−115, 174, 257)-Net in Base 3 — Upper bound on s

There is no (59, 174, 258)-net in base 3, because

1 times m-reduction [i] would yield (59, 173, 258)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40387 199422 523284 348764 556840 088649 119822 517702 222566 676991 706564 066188 680678 252885 > 3¹⁷³ [i]