MinT - Best Known (66, 66+129, s)-Nets in Base 3

Best Known (66, 66+129, s)-Nets in Base 3

(66, 66+129, 48)-Net over F₃ — Constructive and digital

Digital (66, 195, 48)-net over F₃, using

t-expansion [i] based on digital (45, 195, 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+129, 64)-Net over F₃ — Digital

Digital (66, 195, 64)-net over F₃, using

t-expansion [i] based on digital (49, 195, 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+129, 213)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 195, 214)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁵, 214, F₃, 129) (dual of [214, 19, 130]-code), but
- residual code [i] would yield OA(3⁶⁶, 84, S₃, 43), but
  - the linear programming bound shows that MÂ â‰¥ 3447 047885 971598 137581 249186 171087 026769 / 88 665115 > 3⁶⁶ [i]

(66, 66+129, 285)-Net in Base 3 — Upper bound on s

There is no (66, 195, 286)-net in base 3, because

1 times m-reduction [i] would yield (66, 194, 286)-net in base 3, but
- 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]