MinT - Best Known (40−21, 40, s)-Nets in Base 64

Best Known (40−21, 40, s)-Nets in Base 64

(40−21, 40, 257)-Net over F₆₄ — Constructive and digital

Digital (19, 40, 257)-net over F₆₄, using

1 times m-reduction [i] based on digital (19, 41, 257)-net over F₆₄, 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. digital (7, 29, 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]

(40−21, 40, 386)-Net in Base 64 — Constructive

(19, 40, 386)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (2, 12, 129)-net in base 64, using
  - 2 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
    - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. (7, 28, 257)-net in base 64, using
  - base change [i] based on digital (0, 21, 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]

(40−21, 40, 633)-Net over F₆₄ — Digital

Digital (19, 40, 633)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁰, 633, F₆₄, 21) (dual of [633, 593, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁴⁰, 819, F₆₄, 21) (dual of [819, 779, 22]-code), using
  - the BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â {−1,0,…,19}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(40−21, 40, 795673)-Net in Base 64 — Upper bound on s

There is no (19, 40, 795674)-net in base 64, because

1 times m-reduction [i] would yield (19, 39, 795674)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 27607 021892 100896 690908 800976 217992 948625 658844 913591 858952 707001 194776 > 64³⁹ [i]