MinT - Best Known (89−43, 89, s)-Nets in Base 64

Best Known (89−43, 89, s)-Nets in Base 64

(89−43, 89, 513)-Net over F₆₄ — Constructive and digital

Digital (46, 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]

(89−43, 89, 545)-Net in Base 64 — Constructive

(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]

(89−43, 89, 2055)-Net over F₆₄ — Digital

Digital (46, 89, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁹, 2055, F₆₄, 2, 43) (dual of [(2055, 2), 4021, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁸⁹, 4110, F₆₄, 43) (dual of [4110, 4021, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(37) [i] based on
    1. linear OA(64⁸⁵, 4096, F₆₄, 43) (dual of [4096, 4011, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. linear OA(64⁷⁵, 4096, F₆₄, 38) (dual of [4096, 4021, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    3. linear OA(64⁴, 14, F₆₄, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁴, 64, F₆₄, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
        Reedâ€“Solomon code RS(60,64) [i]

(89−43, 89, 5103937)-Net in Base 64 — Upper bound on s

There is no (46, 89, 5103938)-net in base 64, because

1 times m-reduction [i] would yield (46, 88, 5103938)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 878 697285 568975 340690 765232 068815 922818 302620 095484 472140 225354 403385 305507 453102 229408 706046 851313 690388 181216 977055 050329 344791 899625 419339 264871 632205 096720 > 64⁸⁸ [i]