MinT - Best Known (38, 52, s)-Nets in Base 81

Best Known (38, 52, s)-Nets in Base 81

(38, 52, 76166)-Net over F₈₁ — Constructive and digital

Digital (38, 52, 76166)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 246)-net over F₈₁, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 82)-net over F₈₁, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 3, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
    3. digital (0, 7, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (26, 40, 75920)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁰, 75920, F₈₁, 14, 14) (dual of [(75920, 14), 1062840, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(81⁴⁰, 531440, F₈₁, 14) (dual of [531440, 531400, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴⁰, 531441, F₈₁, 14) (dual of [531441, 531401, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(38, 52, 3049700)-Net over F₈₁ — Digital

Digital (38, 52, 3049700)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(38, 52, large)-Net in Base 81 — Upper bound on s

There is no (38, 52, large)-net in base 81, because

12 times m-reduction [i] would yield (38, 40, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]