MinT - Best Known (50, 50+31, s)-Nets in Base 64

Best Known (50, 50+31, s)-Nets in Base 64

(50, 50+31, 690)-Net over F₆₄ — Constructive and digital

Digital (50, 81, 690)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 22, 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, 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]

(50, 50+31, 1093)-Net in Base 64 — Constructive

(50, 81, 1093)-net in base 64, using

64⁴ times duplication [i] based on (46, 77, 1093)-net in base 64, using
- base change [i] based on digital (35, 66, 1093)-net over F₁₂₈, using
  - 128² times duplication [i] based on digital (33, 64, 1093)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁶⁴, 1093, F₁₂₈, 31, 31) (dual of [(1093, 31), 33819, 32]-NRT-code), using
      - OOA 15-folding and stacking with additional row [i] based on linear OA(128⁶⁴, 16396, F₁₂₈, 31) (dual of [16396, 16332, 32]-code), using
        constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
        linear OA(128⁶¹, 16385, F₁₂₈, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (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(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]

(50, 50+31, 14407)-Net over F₆₄ — Digital

Digital (50, 81, 14407)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+31, large)-Net in Base 64 — Upper bound on s

There is no (50, 81, large)-net in base 64, because

29 times m-reduction [i] would yield (50, 52, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (50, 50+31, s)-Nets in Base 64

(50, 50+31, 690)-Net over F64 — Constructive and digital

(50, 50+31, 1093)-Net in Base 64 — Constructive

(50, 50+31, 14407)-Net over F64 — Digital

(50, 50+31, large)-Net in Base 64 — Upper bound on s

(50, 50+31, 690)-Net over F₆₄ — Constructive and digital

(50, 50+31, 14407)-Net over F₆₄ — Digital