MinT - Best Known (45, 45+17, s)-Nets in Base 81

Best Known (45, 45+17, s)-Nets in Base 81

(45, 45+17, 66612)-Net over F₈₁ — Constructive and digital

Digital (45, 62, 66612)-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 (32, 49, 66430)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁹, 66430, F₈₁, 17, 17) (dual of [(66430, 17), 1129261, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(81⁴⁹, 531441, F₈₁, 17) (dual of [531441, 531392, 18]-code), using
      - an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(45, 45+17, 2112661)-Net over F₈₁ — Digital

Digital (45, 62, 2112661)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+17, large)-Net in Base 81 — Upper bound on s

There is no (45, 62, large)-net in base 81, because

15 times m-reduction [i] would yield (45, 47, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]