MinT - Best Known (46−24, 46, s)-Nets in Base 64

Best Known (46−24, 46, s)-Nets in Base 64

(46−24, 46, 281)-Net over F₆₄ — Constructive and digital

Digital (22, 46, 281)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 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, 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]

(46−24, 46, 386)-Net in Base 64 — Constructive

(22, 46, 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. (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]

(46−24, 46, 701)-Net over F₆₄ — Digital

Digital (22, 46, 701)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁶, 701, F₆₄, 24) (dual of [701, 655, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁴⁶, 820, F₆₄, 24) (dual of [820, 774, 25]-code), using
  - an extension C_e(23) of the narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(46−24, 46, 704218)-Net in Base 64 — Upper bound on s

There is no (22, 46, 704219)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 121418 426476 064550 358201 972134 632659 415217 068717 542593 321537 115816 922526 551068 292980 > 64⁴⁶ [i]