MinT - Best Known (23, 48, s)-Nets in Base 64

Best Known (23, 48, s)-Nets in Base 64

(23, 48, 281)-Net over F₆₄ — Constructive and digital

Digital (23, 48, 281)-net over F₆₄, using

1 times m-reduction [i] based on digital (23, 49, 281)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 16, 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, 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]

(23, 48, 386)-Net in Base 64 — Constructive

(23, 48, 386)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (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. (9, 34, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
    - base change [i] based on digital (0, 27, 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]

(23, 48, 724)-Net over F₆₄ — Digital

Digital (23, 48, 724)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁸, 724, F₆₄, 25) (dual of [724, 676, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁴⁸, 820, F₆₄, 25) (dual of [820, 772, 26]-code), using
  - an extension C_e(24) of the narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(23, 48, 995917)-Net in Base 64 — Upper bound on s

There is no (23, 48, 995918)-net in base 64, because

1 times m-reduction [i] would yield (23, 47, 995918)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7 770740 180573 442133 379263 911379 488598 311191 062287 934027 403066 747221 070561 525894 731285 > 64⁴⁷ [i]