MinT - Best Known (64, 64+25, s)-Nets in Base 8

Best Known (64, 64+25, s)-Nets in Base 8

(64, 64+25, 484)-Net over F₈ — Constructive and digital

Digital (64, 89, 484)-net over F₈, using

8¹ times duplication [i] based on digital (63, 88, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (12, 24, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 12, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (39, 64, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 32, 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]

(64, 64+25, 576)-Net in Base 8 — Constructive

(64, 89, 576)-net in base 8, using

7 times m-reduction [i] based on (64, 96, 576)-net in base 8, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 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]

(64, 64+25, 3829)-Net over F₈ — Digital

Digital (64, 89, 3829)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁹, 3829, F₈, 25) (dual of [3829, 3740, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁹, 4097, F₈, 25) (dual of [4097, 4008, 26]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(64, 64+25, 3169000)-Net in Base 8 — Upper bound on s

There is no (64, 89, 3169001)-net in base 8, because

1 times m-reduction [i] would yield (64, 88, 3169001)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 642845 812393 080860 236228 287091 096265 742793 966317 248087 376102 456507 604871 571736 > 8⁸⁸ [i]