MinT - Best Known (77−32, 77, s)-Nets in Base 64

Best Known (77−32, 77, s)-Nets in Base 64

(77−32, 77, 593)-Net over F₆₄ — Constructive and digital

Digital (45, 77, 593)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (28, 60, 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]

(77−32, 77, 1024)-Net in Base 64 — Constructive

(45, 77, 1024)-net in base 64, using

t-expansion [i] based on (44, 77, 1024)-net in base 64, using
- base change [i] based on digital (33, 66, 1024)-net over F₁₂₈, using
  - 128¹ times duplication [i] based on digital (32, 65, 1024)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁶⁵, 1024, F₁₂₈, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
      - OOA 16-folding and stacking with additional row [i] based on linear OA(128⁶⁵, 16385, F₁₂₈, 33) (dual of [16385, 16320, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(77−32, 77, 6055)-Net over F₆₄ — Digital

Digital (45, 77, 6055)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−32, 77, large)-Net in Base 64 — Upper bound on s

There is no (45, 77, large)-net in base 64, because

30 times m-reduction [i] would yield (45, 47, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]