MinT - Best Known (42−22, 42, s)-Nets in Base 64

Best Known (42−22, 42, s)-Nets in Base 64

(42−22, 42, 257)-Net over F₆₄ — Constructive and digital

Digital (20, 42, 257)-net over F₆₄, using

2 times m-reduction [i] based on digital (20, 44, 257)-net over F₆₄, 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. digital (7, 31, 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]

(42−22, 42, 337)-Net in Base 64 — Constructive

(20, 42, 337)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 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, 30, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (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]

(42−22, 42, 682)-Net over F₆₄ — Digital

Digital (20, 42, 682)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴², 682, F₆₄, 2, 22) (dual of [(682, 2), 1322, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁴², 1364, F₆₄, 22) (dual of [1364, 1322, 23]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁴², 1365, F₆₄, 22) (dual of [1365, 1323, 23]-code), using
    - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,69,137,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(42−22, 42, 613752)-Net in Base 64 — Upper bound on s

There is no (20, 42, 613753)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7237 032291 906000 511790 985972 806321 603423 178706 684594 769061 486167 676114 262480 > 64⁴² [i]