MinT - Best Known (65, 65+130, s)-Nets in Base 3

Best Known (65, 65+130, s)-Nets in Base 3

(65, 65+130, 48)-Net over F₃ — Constructive and digital

Digital (65, 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]

(65, 65+130, 64)-Net over F₃ — Digital

Digital (65, 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]

(65, 65+130, 208)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 195, 209)-net over F₃, because

1 times m-reduction [i] would yield digital (65, 194, 209)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁴, 209, F₃, 129) (dual of [209, 15, 130]-code), but
  - residual code [i] would yield OA(3⁶⁵, 79, S₃, 43), but
    - the linear programming bound shows that MÂ â‰¥ 172 665163 578928 812417 146840 028215 778171 / 14 734291 > 3⁶⁵ [i]

(65, 65+130, 278)-Net in Base 3 — Upper bound on s

There is no (65, 195, 279)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1264 268860 993259 039608 715225 896928 836949 172393 015770 870525 384254 683502 721144 698708 999901 956527 > 3¹⁹⁵ [i]