MinT - Best Known (48, 48+18, s)-Nets in Base 81

Best Known (48, 48+18, s)-Nets in Base 81

(48, 48+18, 59231)-Net over F₈₁ — Constructive and digital

Digital (48, 66, 59231)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 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, 10, 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 (34, 52, 59049)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵², 59049, F₈₁, 18, 18) (dual of [(59049, 18), 1062830, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(81⁵², 531441, F₈₁, 18) (dual of [531441, 531389, 19]-code), using
      - an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(48, 48+18, 2302869)-Net over F₈₁ — Digital

Digital (48, 66, 2302869)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(48, 48+18, large)-Net in Base 81 — Upper bound on s

There is no (48, 66, large)-net in base 81, because

16 times m-reduction [i] would yield (48, 50, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]