MinT - Best Known (32−17, 32, s)-Nets in Base 64

Best Known (32−17, 32, s)-Nets in Base 64

(32−17, 32, 242)-Net over F₆₄ — Constructive and digital

Digital (15, 32, 242)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 24, 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]

(32−17, 32, 337)-Net in Base 64 — Constructive

(15, 32, 337)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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. (6, 23, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
    - base change [i] based on digital (0, 18, 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]

(32−17, 32, 545)-Net over F₆₄ — Digital

Digital (15, 32, 545)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³², 545, F₆₄, 17) (dual of [545, 513, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(64³², 819, F₆₄, 17) (dual of [819, 787, 18]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(32−17, 32, 596065)-Net in Base 64 — Upper bound on s

There is no (15, 32, 596066)-net in base 64, because

1 times m-reduction [i] would yield (15, 31, 596066)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 98 079854 707856 271371 373927 428261 961384 145809 855876 132532 > 64³¹ [i]