MinT - Best Known (71−28, 71, s)-Nets in Base 81

Best Known (71−28, 71, s)-Nets in Base 81

(71−28, 71, 740)-Net over F₈₁ — Constructive and digital

Digital (43, 71, 740)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (16, 44, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(71−28, 71, 14267)-Net over F₈₁ — Digital

Digital (43, 71, 14267)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−28, 71, large)-Net in Base 81 — Upper bound on s

There is no (43, 71, large)-net in base 81, because

26 times m-reduction [i] would yield (43, 45, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]