MinT - Best Known (72−28, 72, s)-Nets in Base 64

Best Known (72−28, 72, s)-Nets in Base 64

(72−28, 72, 593)-Net over F₆₄ — Constructive and digital

Digital (44, 72, 593)-net over F₆₄, using

2 times m-reduction [i] based on digital (44, 74, 593)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. 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, 58, 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]

(72−28, 72, 1171)-Net in Base 64 — Constructive

(44, 72, 1171)-net in base 64, using

64² times duplication [i] based on (42, 70, 1171)-net in base 64, using
- t-expansion [i] based on (41, 70, 1171)-net in base 64, using
  - base change [i] based on digital (31, 60, 1171)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁶⁰, 1171, F₁₂₈, 29, 29) (dual of [(1171, 29), 33899, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(128⁶⁰, 16395, F₁₂₈, 29) (dual of [16395, 16335, 30]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁶⁰, 16396, F₁₂₈, 29) (dual of [16396, 16336, 30]-code), using
        constructionÂ X applied to C([0,14]) âŠ‚ C([0,12]) [i] based on
        linear OA(128⁵⁷, 16385, F₁₂₈, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
        linear OA(128⁴⁹, 16385, F₁₂₈, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(128³, 11, F₁₂₈, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(72−28, 72, 11378)-Net over F₆₄ — Digital

Digital (44, 72, 11378)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−28, 72, large)-Net in Base 64 — Upper bound on s

There is no (44, 72, large)-net in base 64, because

26 times m-reduction [i] would yield (44, 46, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (72−28, 72, s)-Nets in Base 64

(72−28, 72, 593)-Net over F64 — Constructive and digital

(72−28, 72, 1171)-Net in Base 64 — Constructive

(72−28, 72, 11378)-Net over F64 — Digital

(72−28, 72, large)-Net in Base 64 — Upper bound on s

(72−28, 72, 593)-Net over F₆₄ — Constructive and digital

(72−28, 72, 11378)-Net over F₆₄ — Digital