MinT - Best Known (64, 91, s)-Nets in Base 27

Best Known (64, 91, s)-Nets in Base 27

(64, 91, 1541)-Net over F₂₇ — Constructive and digital

Digital (64, 91, 1541)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
2. digital (51, 78, 1513)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷⁸, 1513, F₂₇, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(27⁷⁸, 19670, F₂₇, 27) (dual of [19670, 19592, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁷⁸, 19682, F₂₇, 27) (dual of [19682, 19604, 28]-code), using
        1 times truncation [i] based on linear OA(27⁷⁹, 19683, F₂₇, 28) (dual of [19683, 19604, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(64, 91, 41517)-Net over F₂₇ — Digital

Digital (64, 91, 41517)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 91, large)-Net in Base 27 — Upper bound on s

There is no (64, 91, large)-net in base 27, because

25 times m-reduction [i] would yield (64, 66, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]