MinT - Best Known (43−14, 43, s)-Nets in Base 3

Best Known (43−14, 43, s)-Nets in Base 3

(43−14, 43, 84)-Net over F₃ — Constructive and digital

Digital (29, 43, 84)-net over F₃, using

3¹ times duplication [i] based on digital (28, 42, 84)-net over F₃, using
- trace code for nets [i] based on digital (0, 14, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]

(43−14, 43, 115)-Net over F₃ — Digital

Digital (29, 43, 115)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴³, 115, F₃, 14) (dual of [115, 72, 15]-code), using
- 1 times truncation [i] based on linear OA(3⁴⁴, 116, F₃, 15) (dual of [116, 72, 16]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁶, 58, F₃, 7) (dual of [58, 42, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]
    2. linear OA(3²⁸, 58, F₃, 15) (dual of [58, 30, 16]-code), using
      - 2 times truncation [i] based on linear OA(3³⁰, 60, F₃, 17) (dual of [60, 30, 18]-code), using
        extended quadratic residue code Q_e(60,3) [i]

(43−14, 43, 1434)-Net in Base 3 — Upper bound on s

There is no (29, 43, 1435)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 328 440000 620500 523235 > 3⁴³ [i]