MinT - Best Known (26, 55, s)-Nets in Base 64

Best Known (26, 55, s)-Nets in Base 64

(26, 55, 305)-Net over F₆₄ — Constructive and digital

Digital (26, 55, 305)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]
2. digital (7, 36, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(26, 55, 354)-Net in Base 64 — Constructive

(26, 55, 354)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (5, 19, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
    - base change [i] based on digital (0, 15, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (7, 36, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(26, 55, 698)-Net over F₆₄ — Digital

Digital (26, 55, 698)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁵, 698, F₆₄, 29) (dual of [698, 643, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵⁵, 819, F₆₄, 29) (dual of [819, 764, 30]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(26, 55, 888808)-Net in Base 64 — Upper bound on s

There is no (26, 55, 888809)-net in base 64, because

1 times m-reduction [i] would yield (26, 54, 888809)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 34 176181 327643 392890 996939 812009 575680 103580 102869 790055 454334 982892 447367 397400 941577 352525 344720 > 64⁵⁴ [i]