MinT - Best Known (59−13, 59, s)-Nets in Base 81

Best Known (59−13, 59, s)-Nets in Base 81

(59−13, 59, 1398282)-Net over F₈₁ — Constructive and digital

Digital (46, 59, 1398282)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 182)-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 (1, 7, 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 (36, 49, 1398100)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁹, 1398100, F₈₁, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(81⁴⁹, 8388601, F₈₁, 13) (dual of [8388601, 8388552, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(59−13, 59, large)-Net over F₈₁ — Digital

Digital (46, 59, large)-net over F₈₁, using

t-expansion [i] based on digital (45, 59, large)-net over F₈₁, using
- 2 times m-reduction [i] based on digital (45, 61, large)-net over F₈₁, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶¹, large, F₈₁, 16) (dual of [large, large−61, 17]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(59−13, 59, large)-Net in Base 81 — Upper bound on s

There is no (46, 59, large)-net in base 81, because

11 times m-reduction [i] would yield (46, 48, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (59−13, 59, s)-Nets in Base 81

(59−13, 59, 1398282)-Net over F81 — Constructive and digital

(59−13, 59, large)-Net over F81 — Digital

(59−13, 59, large)-Net in Base 81 — Upper bound on s

(59−13, 59, 1398282)-Net over F₈₁ — Constructive and digital

(59−13, 59, large)-Net over F₈₁ — Digital