MinT - Best Known (66, 66+128, s)-Nets in Base 3

Best Known (66, 66+128, s)-Nets in Base 3

(66, 66+128, 48)-Net over F₃ — Constructive and digital

Digital (66, 194, 48)-net over F₃, using

t-expansion [i] based on digital (45, 194, 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]

(66, 66+128, 64)-Net over F₃ — Digital

Digital (66, 194, 64)-net over F₃, using

t-expansion [i] based on digital (49, 194, 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]

(66, 66+128, 220)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 194, 221)-net over F₃, because

2 times m-reduction [i] would yield digital (66, 192, 221)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹², 221, F₃, 126) (dual of [221, 29, 127]-code), but
  - residual code [i] would yield OA(3⁶⁶, 94, S₃, 42), but
    - the linear programming bound shows that MÂ â‰¥ 1 747070 674352 666159 830908 506217 598892 875513 970337 / 51301 241878 906250 > 3⁶⁶ [i]

(66, 66+128, 285)-Net in Base 3 — Upper bound on s

There is no (66, 194, 286)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 387 710736 694020 286492 877139 192130 737324 116925 768874 815411 260349 142646 882633 107386 457809 735297 > 3¹⁹⁴ [i]