MinT - Best Known (200−135, 200, s)-Nets in Base 3

Best Known (200−135, 200, s)-Nets in Base 3

(200−135, 200, 48)-Net over F₃ — Constructive and digital

Digital (65, 200, 48)-net over F₃, using

t-expansion [i] based on digital (45, 200, 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]

(200−135, 200, 64)-Net over F₃ — Digital

Digital (65, 200, 64)-net over F₃, using

t-expansion [i] based on digital (49, 200, 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]

(200−135, 200, 204)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 200, 205)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁰⁰, 205, F₃, 135) (dual of [205, 5, 136]-code), but
- Griesmer bound [i]

(200−135, 200, 275)-Net in Base 3 — Upper bound on s

There is no (65, 200, 276)-net in base 3, because

1 times m-reduction [i] would yield (65, 199, 276)-net in base 3, but
- 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]