MinT - Best Known (28, 28+16, s)-Nets in Base 81

Best Known (28, 28+16, s)-Nets in Base 81

(28, 28+16, 1002)-Net over F₈₁ — Constructive and digital

Digital (28, 44, 1002)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (5, 13, 182)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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 (1, 9, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
2. digital (15, 31, 820)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³¹, 820, F₈₁, 16, 16) (dual of [(820, 16), 13089, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(81³¹, 6560, F₈₁, 16) (dual of [6560, 6529, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(81³¹, 6561, F₈₁, 16) (dual of [6561, 6530, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(28, 28+16, 31842)-Net over F₈₁ — Digital

Digital (28, 44, 31842)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 28+16, large)-Net in Base 81 — Upper bound on s

There is no (28, 44, large)-net in base 81, because

14 times m-reduction [i] would yield (28, 30, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]