MinT - Best Known (89−42, 89, s)-Nets in Base 64

Best Known (89−42, 89, s)-Nets in Base 64

(89−42, 89, 513)-Net over F₆₄ — Constructive and digital

Digital (47, 89, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 89, 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]

(89−42, 89, 545)-Net in Base 64 — Constructive

(47, 89, 545)-net in base 64, using

t-expansion [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]

(89−42, 89, 2336)-Net over F₆₄ — Digital

Digital (47, 89, 2336)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁹, 2336, F₆₄, 42) (dual of [2336, 2247, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁸⁹, 4116, F₆₄, 42) (dual of [4116, 4027, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(34) [i] based on
    1. linear OA(64⁸³, 4096, F₆₄, 42) (dual of [4096, 4013, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. linear OA(64⁶⁹, 4096, F₆₄, 35) (dual of [4096, 4027, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(64⁶, 20, F₆₄, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁶, 64, F₆₄, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
        Reedâ€“Solomon code RS(58,64) [i]

(89−42, 89, 6221771)-Net in Base 64 — Upper bound on s

There is no (47, 89, 6221772)-net in base 64, because

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]