MinT - Best Known (90−33, 90, s)-Nets in Base 64

Best Known (90−33, 90, s)-Nets in Base 64

(90−33, 90, 708)-Net over F₆₄ — Constructive and digital

Digital (57, 90, 708)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 29, 195)-net over F₆₄, using
  - 1 times m-reduction [i] based on digital (13, 30, 195)-net over F₆₄, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 5, 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 (0, 8, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
      3. digital (0, 17, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (28, 61, 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]

(90−33, 90, 4096)-Net in Base 64 — Constructive

(57, 90, 4096)-net in base 64, using

64² times duplication [i] based on (55, 88, 4096)-net in base 64, using
- base change [i] based on digital (33, 66, 4096)-net over F₂₅₆, using
  - 256¹ times duplication [i] based on digital (32, 65, 4096)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶⁵, 4096, F₂₅₆, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
      - OOA 16-folding and stacking with additional row [i] based on linear OA(256⁶⁵, 65537, F₂₅₆, 33) (dual of [65537, 65472, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(90−33, 90, 24418)-Net over F₆₄ — Digital

Digital (57, 90, 24418)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−33, 90, large)-Net in Base 64 — Upper bound on s

There is no (57, 90, large)-net in base 64, because

31 times m-reduction [i] would yield (57, 59, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]