MinT - Best Known (17, 36, s)-Nets in Base 64

Best Known (17, 36, s)-Nets in Base 64

(17, 36, 257)-Net over F₆₄ — Constructive and digital

Digital (17, 36, 257)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (7, 26, 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]

(17, 36, 337)-Net in Base 64 — Constructive

(17, 36, 337)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. (7, 26, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
    - base change [i] based on digital (0, 21, 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]

(17, 36, 589)-Net over F₆₄ — Digital

Digital (17, 36, 589)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁶, 589, F₆₄, 19) (dual of [589, 553, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁶, 819, F₆₄, 19) (dual of [819, 783, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(17, 36, 695731)-Net in Base 64 — Upper bound on s

There is no (17, 36, 695732)-net in base 64, because

1 times m-reduction [i] would yield (17, 35, 695732)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1645 519368 427483 255773 576799 749560 904829 687507 077075 436769 996338 > 64³⁵ [i]