MinT - Best Known (89−34, 89, s)-Nets in Base 64

Best Known (89−34, 89, s)-Nets in Base 64

(89−34, 89, 690)-Net over F₆₄ — Constructive and digital

Digital (55, 89, 690)-net over F₆₄, using

t-expansion [i] based on digital (53, 89, 690)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 25, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]
  2. digital (28, 64, 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]

(89−34, 89, 965)-Net in Base 64 — Constructive

(55, 89, 965)-net in base 64, using

t-expansion [i] based on (54, 89, 965)-net in base 64, using
- net defined by OOA [i] based on OOA(64⁸⁹, 965, S₆₄, 35, 35), using
  - OOA 17-folding and stacking with additional row [i] based on OA(64⁸⁹, 16406, S₆₄, 35), using
    - discarding factors based on OA(64⁸⁹, 16408, S₆₄, 35), using
      - discarding parts of the base [i] based on linear OA(128⁷⁶, 16408, F₁₂₈, 35) (dual of [16408, 16332, 36]-code), using
        constructionÂ X applied to C([0,17]) âŠ‚ C([0,13]) [i] based on
        linear OA(128⁶⁹, 16385, F₁₂₈, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
        linear OA(128⁵³, 16385, F₁₂₈, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(128⁷, 23, F₁₂₈, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁷, 128, F₁₂₈, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
        Reedâ€“Solomon code RS(121,128) [i]

(89−34, 89, 15548)-Net over F₆₄ — Digital

Digital (55, 89, 15548)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−34, 89, large)-Net in Base 64 — Upper bound on s

There is no (55, 89, large)-net in base 64, because

32 times m-reduction [i] would yield (55, 57, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]