MinT - Best Known (24, 50, s)-Nets in Base 64

Best Known (24, 50, s)-Nets in Base 64

(24, 50, 281)-Net over F₆₄ — Constructive and digital

Digital (24, 50, 281)-net over F₆₄, using

2 times m-reduction [i] based on digital (24, 52, 281)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 17, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (7, 35, 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]

(24, 50, 338)-Net in Base 64 — Constructive

(24, 50, 338)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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. (10, 36, 258)-net in base 64, using
  - base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(24, 50, 747)-Net over F₆₄ — Digital

Digital (24, 50, 747)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁰, 747, F₆₄, 26) (dual of [747, 697, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵⁰, 820, F₆₄, 26) (dual of [820, 770, 27]-code), using
  - an extension C_e(25) of the narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(24, 50, 795991)-Net in Base 64 — Upper bound on s

There is no (24, 50, 795992)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 037060 769725 648532 788748 250862 354854 828351 327343 977844 089826 593576 925361 275497 757857 495830 > 64⁵⁰ [i]