MinT - Best Known (58, 58+111, s)-Nets in Base 3

Best Known (58, 58+111, s)-Nets in Base 3

(58, 58+111, 48)-Net over F₃ — Constructive and digital

Digital (58, 169, 48)-net over F₃, using

t-expansion [i] based on digital (45, 169, 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]

(58, 58+111, 64)-Net over F₃ — Digital

Digital (58, 169, 64)-net over F₃, using

t-expansion [i] based on digital (49, 169, 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]

(58, 58+111, 199)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 169, 200)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁹, 200, F₃, 111) (dual of [200, 31, 112]-code), but
- residual code [i] would yield OA(3⁵⁸, 88, S₃, 37), but
  - the linear programming bound shows that MÂ â‰¥ 161 188690 179521 564774 743221 611934 175250 646363 / 29813 529247 923155 > 3⁵⁸ [i]

(58, 58+111, 255)-Net in Base 3 — Upper bound on s

There is no (58, 169, 256)-net in base 3, because

1 times m-reduction [i] would yield (58, 168, 256)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 170 323088 348461 575979 709829 588714 133952 738594 471325 273502 873684 035332 682982 575105 > 3¹⁶⁸ [i]