MinT - Best Known (27, 27+30, s)-Nets in Base 64

Best Known (27, 27+30, s)-Nets in Base 64

(27, 27+30, 305)-Net over F₆₄ — Constructive and digital

Digital (27, 57, 305)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 20, 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, 37, 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]

(27, 27+30, 354)-Net in Base 64 — Constructive

(27, 57, 354)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (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, 37, 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]

(27, 27+30, 722)-Net over F₆₄ — Digital

Digital (27, 57, 722)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁷, 722, F₆₄, 30) (dual of [722, 665, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵⁷, 819, F₆₄, 30) (dual of [819, 762, 31]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(27, 27+30, 744570)-Net in Base 64 — Upper bound on s

There is no (27, 57, 744571)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8 959086 168924 072379 492352 076307 512093 938272 050891 737638 009027 080751 971096 965277 771862 678443 809191 541776 > 64⁵⁷ [i]