MinT - Best Known (83−32, 83, s)-Nets in Base 64

Best Known (83−32, 83, s)-Nets in Base 64

(83−32, 83, 690)-Net over F₆₄ — Constructive and digital

Digital (51, 83, 690)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 23, 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, 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]

(83−32, 83, 1025)-Net in Base 64 — Constructive

(51, 83, 1025)-net in base 64, using

1 times m-reduction [i] based on (51, 84, 1025)-net in base 64, using
- base change [i] based on digital (39, 72, 1025)-net over F₁₂₈, using
  - 128² times duplication [i] based on digital (37, 70, 1025)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁷⁰, 1025, F₁₂₈, 33, 33) (dual of [(1025, 33), 33755, 34]-NRT-code), using
      - OOA 16-folding and stacking with additional row [i] based on linear OA(128⁷⁰, 16401, F₁₂₈, 33) (dual of [16401, 16331, 34]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁷⁰, 16402, F₁₂₈, 33) (dual of [16402, 16332, 34]-code), using
        constructionÂ X applied to C([0,16]) âŠ‚ C([0,13]) [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]
        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(128⁵, 17, F₁₂₈, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁵, 128, F₁₂₈, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
        Reedâ€“Solomon code RS(123,128) [i]

(83−32, 83, 13524)-Net over F₆₄ — Digital

Digital (51, 83, 13524)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−32, 83, large)-Net in Base 64 — Upper bound on s

There is no (51, 83, large)-net in base 64, because

30 times m-reduction [i] would yield (51, 53, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (83−32, 83, s)-Nets in Base 64

(83−32, 83, 690)-Net over F64 — Constructive and digital

(83−32, 83, 1025)-Net in Base 64 — Constructive

(83−32, 83, 13524)-Net over F64 — Digital

(83−32, 83, large)-Net in Base 64 — Upper bound on s

(83−32, 83, 690)-Net over F₆₄ — Constructive and digital

(83−32, 83, 13524)-Net over F₆₄ — Digital