MinT - Best Known (62−22, 62, s)-Nets in Base 81

Best Known (62−22, 62, s)-Nets in Base 81

(62−22, 62, 842)-Net over F₈₁ — Constructive and digital

Digital (40, 62, 842)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (8, 19, 246)-net over F₈₁, using
  - generalized (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 (0, 5, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
    3. digital (0, 11, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (21, 43, 596)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴³, 596, F₈₁, 22, 22) (dual of [(596, 22), 13069, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(81⁴³, 6556, F₈₁, 22) (dual of [6556, 6513, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴³, 6561, F₈₁, 22) (dual of [6561, 6518, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(62−22, 62, 46782)-Net over F₈₁ — Digital

Digital (40, 62, 46782)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−22, 62, large)-Net in Base 81 — Upper bound on s

There is no (40, 62, large)-net in base 81, because

20 times m-reduction [i] would yield (40, 42, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]