MinT - Best Known (65, 65+134, s)-Nets in Base 3

Best Known (65, 65+134, s)-Nets in Base 3

(65, 65+134, 48)-Net over F₃ — Constructive and digital

Digital (65, 199, 48)-net over F₃, using

t-expansion [i] based on digital (45, 199, 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+134, 64)-Net over F₃ — Digital

Digital (65, 199, 64)-net over F₃, using

t-expansion [i] based on digital (49, 199, 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+134, 205)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 199, 206)-net over F₃, because

2 times m-reduction [i] would yield digital (65, 197, 206)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁷, 206, F₃, 132) (dual of [206, 9, 133]-code), but
  - residual code [i] would yield linear OA(3⁶⁵, 73, F₃, 44) (dual of [73, 8, 45]-code), but
    - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(65, 65+134, 275)-Net in Base 3 — Upper bound on s

There is no (65, 199, 276)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 101592 678718 921374 899448 082942 419649 117492 914028 838518 341340 756530 653344 568855 715936 435103 836273 > 3¹⁹⁹ [i]