MinT - Best Known (34, 34+20, s)-Nets in Base 81

Best Known (34, 34+20, s)-Nets in Base 81

(34, 34+20, 820)-Net over F₈₁ — Constructive and digital

Digital (34, 54, 820)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 164)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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]
    2. digital (0, 10, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (19, 39, 656)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³⁹, 656, F₈₁, 20, 20) (dual of [(656, 20), 13081, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(81³⁹, 6560, F₈₁, 20) (dual of [6560, 6521, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(81³⁹, 6561, F₈₁, 20) (dual of [6561, 6522, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(34, 34+20, 26327)-Net over F₈₁ — Digital

Digital (34, 54, 26327)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+20, large)-Net in Base 81 — Upper bound on s

There is no (34, 54, large)-net in base 81, because

18 times m-reduction [i] would yield (34, 36, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]