MinT - Best Known (21, 45, s)-Nets in Base 64

Best Known (21, 45, s)-Nets in Base 64

(21, 45, 257)-Net over F₆₄ — Constructive and digital

Digital (21, 45, 257)-net over F₆₄, using

2 times m-reduction [i] based on digital (21, 47, 257)-net over F₆₄, 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. digital (7, 33, 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]

(21, 45, 337)-Net in Base 64 — Constructive

(21, 45, 337)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 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. (8, 32, 257)-net in base 64, using
  - base change [i] based on digital (0, 24, 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]

(21, 45, 579)-Net over F₆₄ — Digital

Digital (21, 45, 579)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁵, 579, F₆₄, 24) (dual of [579, 534, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁴⁵, 585, F₆₄, 24) (dual of [585, 540, 25]-code), using
  - the BCH-code C(I) with length 585Â | 64²âˆ’1, defining interval IÂ =Â {1,60,119,…,188}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(21, 45, 497955)-Net in Base 64 — Upper bound on s

There is no (21, 45, 497956)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1897 146288 515144 944278 775346 266826 926492 751261 373034 884272 029569 740224 730138 020120 > 64⁴⁵ [i]