MinT - Best Known (62, 169, s)-Nets in Base 3

Best Known (62, 169, s)-Nets in Base 3

(62, 169, 48)-Net over F₃ — Constructive and digital

Digital (62, 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]

(62, 169, 64)-Net over F₃ — Digital

Digital (62, 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]

(62, 169, 281)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 169, 282)-net over F₃, because

2 times m-reduction [i] would yield digital (62, 167, 282)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁷, 282, F₃, 105) (dual of [282, 115, 106]-code), but
  - residual code [i] would yield OA(3⁶², 176, S₃, 35), but
    - the linear programming bound shows that MÂ â‰¥ 8 776263 770751 612540 193525 834020 135240 880280 270100 948539 296875 / 21 936508 659210 038625 164225 310787 > 3⁶² [i]

(62, 169, 285)-Net in Base 3 — Upper bound on s

There is no (62, 169, 286)-net in base 3, because

1 times m-reduction [i] would yield (62, 168, 286)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 149 426234 475669 377377 496088 110942 198104 846565 300883 465780 878921 022510 253312 083957 > 3¹⁶⁸ [i]