MinT - Best Known (37−14, 37, s)-Nets in Base 81

Best Known (37−14, 37, s)-Nets in Base 81

(37−14, 37, 1101)-Net over F₈₁ — Constructive and digital

Digital (23, 37, 1101)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 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, 7, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (13, 27, 937)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁷, 937, F₈₁, 14, 14) (dual of [(937, 14), 13091, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(81²⁷, 6559, F₈₁, 14) (dual of [6559, 6532, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(81²⁷, 6561, F₈₁, 14) (dual of [6561, 6534, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(37−14, 37, 19156)-Net over F₈₁ — Digital

Digital (23, 37, 19156)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(37−14, 37, large)-Net in Base 81 — Upper bound on s

There is no (23, 37, large)-net in base 81, because

12 times m-reduction [i] would yield (23, 25, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]