MinT - Best Known (58−16, 58, s)-Nets in Base 81

Best Known (58−16, 58, s)-Nets in Base 81

(58−16, 58, 66594)-Net over F₈₁ — Constructive and digital

Digital (42, 58, 66594)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 164)-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 (0, 8, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (30, 46, 66430)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁶, 66430, F₈₁, 16, 16) (dual of [(66430, 16), 1062834, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(81⁴⁶, 531440, F₈₁, 16) (dual of [531440, 531394, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴⁶, 531441, F₈₁, 16) (dual of [531441, 531395, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(58−16, 58, 1923773)-Net over F₈₁ — Digital

Digital (42, 58, 1923773)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−16, 58, large)-Net in Base 81 — Upper bound on s

There is no (42, 58, large)-net in base 81, because

14 times m-reduction [i] would yield (42, 44, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]