MinT - Best Known (87−31, 87, s)-Nets in Base 64

Best Known (87−31, 87, s)-Nets in Base 64

(87−31, 87, 723)-Net over F₆₄ — Constructive and digital

Digital (56, 87, 723)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 28, 210)-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, 7, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
    3. digital (1, 16, 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, 59, 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]

(87−31, 87, 4369)-Net in Base 64 — Constructive

(56, 87, 4369)-net in base 64, using

64³ times duplication [i] based on (53, 84, 4369)-net in base 64, using
- base change [i] based on digital (32, 63, 4369)-net over F₂₅₆, using
  - 256² times duplication [i] based on digital (30, 61, 4369)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶¹, 4369, F₂₅₆, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
      - OOA 15-folding and stacking with additional row [i] based on linear OA(256⁶¹, 65536, F₂₅₆, 31) (dual of [65536, 65475, 32]-code), using
        an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(87−31, 87, 33080)-Net over F₆₄ — Digital

Digital (56, 87, 33080)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−31, 87, large)-Net in Base 64 — Upper bound on s

There is no (56, 87, large)-net in base 64, because

29 times m-reduction [i] would yield (56, 58, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]