MinT - Best Known (59, 59+111, s)-Nets in Base 3

Best Known (59, 59+111, s)-Nets in Base 3

(59, 59+111, 48)-Net over F₃ — Constructive and digital

Digital (59, 170, 48)-net over F₃, using

t-expansion [i] based on digital (45, 170, 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]

(59, 59+111, 64)-Net over F₃ — Digital

Digital (59, 170, 64)-net over F₃, using

t-expansion [i] based on digital (49, 170, 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]

(59, 59+111, 209)-Net over F₃ — Upper bound on s (digital)

There is no digital (59, 170, 210)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁰, 210, F₃, 111) (dual of [210, 40, 112]-code), but
- residual code [i] would yield OA(3⁵⁹, 98, S₃, 37), but
  - the linear programming bound shows that MÂ â‰¥ 669 706575 926673 602832 294393 126699 440696 899104 765650 881469 270113 504934 632053 / 45175 106642 610955 544989 816374 596571 195177 978293 > 3⁵⁹ [i]

(59, 59+111, 261)-Net in Base 3 — Upper bound on s

There is no (59, 170, 262)-net in base 3, because

1 times m-reduction [i] would yield (59, 169, 262)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 499 899451 724961 048986 163320 991423 533548 816354 190882 153869 356011 862904 567092 178233 > 3¹⁶⁹ [i]