MinT - Best Known (34−13, 34, s)-Nets in Base 81

Best Known (34−13, 34, s)-Nets in Base 81

(34−13, 34, 1257)-Net over F₈₁ — Constructive and digital

Digital (21, 34, 1257)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 164)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (0, 6, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (12, 25, 1093)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁵, 1093, F₈₁, 13, 13) (dual of [(1093, 13), 14184, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(81²⁵, 6559, F₈₁, 13) (dual of [6559, 6534, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(81²⁵, 6561, F₈₁, 13) (dual of [6561, 6536, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(34−13, 34, 16897)-Net over F₈₁ — Digital

Digital (21, 34, 16897)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(34−13, 34, large)-Net in Base 81 — Upper bound on s

There is no (21, 34, large)-net in base 81, because

11 times m-reduction [i] would yield (21, 23, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]