MinT - Best Known (24, 52, s)-Nets in Base 64

Best Known (24, 52, s)-Nets in Base 64

(24, 52, 281)-Net over F₆₄ — Constructive and digital

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, 52, 322)-Net in Base 64 — Constructive

(24, 52, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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. (10, 38, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
    - base change [i] based on digital (0, 30, 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]

(24, 52, 573)-Net over F₆₄ — Digital

Digital (24, 52, 573)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵², 573, F₆₄, 28) (dual of [573, 521, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵², 585, F₆₄, 28) (dual of [585, 533, 29]-code), using
  - the BCH-code C(I) with length 585Â | 64²âˆ’1, defining interval IÂ =Â {1,90,179,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(24, 52, 490658)-Net in Base 64 — Upper bound on s

There is no (24, 52, 490659)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8343 714030 908717 552695 701249 364116 485534 109872 137991 462972 043052 392287 907170 563358 076978 671800 > 64⁵² [i]