MinT - Best Known (19−8, 19, s)-Nets in Base 3

Best Known (19−8, 19, s)-Nets in Base 3

(19−8, 19, 32)-Net over F₃ — Constructive and digital

Digital (11, 19, 32)-net over F₃, using

1 times m-reduction [i] based on digital (11, 20, 32)-net over F₃, using
- trace code for nets [i] based on digital (1, 10, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]

(19−8, 19, 37)-Net over F₃ — Digital

Digital (11, 19, 37)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹, 37, F₃, 8) (dual of [37, 18, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹, 48, F₃, 8) (dual of [48, 29, 9]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3⁷, 24, F₃, 4) (dual of [24, 17, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷, 26, F₃, 4) (dual of [26, 19, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    2. linear OA(3¹², 24, F₃, 8) (dual of [24, 12, 9]-code), using
      - extended quadratic residue code Q_e(24,3) [i]

(19−8, 19, 200)-Net in Base 3 — Upper bound on s

There is no (11, 19, 201)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1165 387281 > 3¹⁹ [i]