MinT - Best Known (46, 91, s)-Nets in Base 64

Best Known (46, 91, s)-Nets in Base 64

(46, 91, 513)-Net over F₆₄ — Constructive and digital

Digital (46, 91, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 91, 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]

(46, 91, 514)-Net in Base 64 — Constructive

(46, 91, 514)-net in base 64, using

64¹ times duplication [i] based on (45, 90, 514)-net in base 64, using
- (u,Â u+v)-construction [i] based on
  1. (8, 30, 257)-net in base 64, using
    - 2 times m-reduction [i] based on (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]
  2. (15, 60, 257)-net in base 64, using
    - base change [i] based on digital (0, 45, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(46, 91, 1741)-Net over F₆₄ — Digital

Digital (46, 91, 1741)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁹¹, 1741, F₆₄, 2, 45) (dual of [(1741, 2), 3391, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁹¹, 2052, F₆₄, 2, 45) (dual of [(2052, 2), 4013, 46]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁹¹, 4104, F₆₄, 45) (dual of [4104, 4013, 46]-code), using
    - constructionÂ X applied to C_e(44) âŠ‚ C_e(41) [i] based on
      1. linear OA(64⁸⁹, 4096, F₆₄, 45) (dual of [4096, 4007, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      2. 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]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(46, 91, 3519080)-Net in Base 64 — Upper bound on s

There is no (46, 91, 3519081)-net in base 64, because

1 times m-reduction [i] would yield (46, 90, 3519081)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 599140 027082 701157 407562 583348 795524 838256 773164 625638 629632 968549 110278 653147 483190 701797 370226 572187 634077 682941 725699 645490 616734 807514 223154 902191 677359 165944 > 64⁹⁰ [i]