MinT - Best Known (47, 47+43, s)-Nets in Base 64

Best Known (47, 47+43, s)-Nets in Base 64

(47, 47+43, 513)-Net over F₆₄ — Constructive and digital

Digital (47, 90, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 90, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(47, 47+43, 545)-Net in Base 64 — Constructive

(47, 90, 545)-net in base 64, using

64¹ times duplication [i] based on (46, 89, 545)-net in base 64, using
- (u,Â u+v)-construction [i] based on
  1. (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]
  2. (18, 61, 288)-net in base 64, using
    - 2 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
      - base change [i] based on digital (9, 54, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(47, 47+43, 2115)-Net over F₆₄ — Digital

Digital (47, 90, 2115)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁹⁰, 2115, F₆₄, 43) (dual of [2115, 2025, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁹⁰, 4114, F₆₄, 43) (dual of [4114, 4024, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(64⁸⁵, 4097, F₆₄, 43) (dual of [4097, 4012, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(64⁷³, 4097, F₆₄, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    3. linear OA(64⁵, 17, F₆₄, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁵, 64, F₆₄, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
        Reedâ€“Solomon code RS(59,64) [i]

(47, 47+43, 6221771)-Net in Base 64 — Upper bound on s

There is no (47, 90, 6221772)-net in base 64, because

1 times m-reduction [i] would yield (47, 89, 6221772)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 56236 555030 044703 626305 605350 695550 174823 983686 525788 062388 490762 163009 127928 961636 676529 973382 470113 721624 501714 782786 826393 657466 676477 347430 579631 146376 007432 > 64⁸⁹ [i]