MinT - Best Known (62, 185, s)-Nets in Base 3

Best Known (62, 185, s)-Nets in Base 3

(62, 185, 48)-Net over F₃ — Constructive and digital

Digital (62, 185, 48)-net over F₃, using

t-expansion [i] based on digital (45, 185, 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]

(62, 185, 64)-Net over F₃ — Digital

Digital (62, 185, 64)-net over F₃, using

t-expansion [i] based on digital (49, 185, 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]

(62, 185, 199)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 185, 200)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁵, 200, F₃, 123) (dual of [200, 15, 124]-code), but
- residual code [i] would yield OA(3⁶², 76, S₃, 41), but
  - the linear programming bound shows that MÂ â‰¥ 347660 486804 616889 071607 220289 701250 / 818741 > 3⁶² [i]

(62, 185, 267)-Net in Base 3 — Upper bound on s

There is no (62, 185, 268)-net in base 3, because

1 times m-reduction [i] would yield (62, 184, 268)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6589 723799 408262 773152 422961 757289 343095 150698 318332 475608 893182 454611 039518 335271 583481 > 3¹⁸⁴ [i]