MinT - Best Known (50, 50+14, s)-Nets in Base 64

Best Known (50, 50+14, s)-Nets in Base 64

(50, 50+14, 1198516)-Net over F₆₄ — Constructive and digital

Digital (50, 64, 1198516)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 145)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (1, 8, 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 (39, 53, 1198371)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(50, 50+14, 1198629)-Net in Base 64 — Constructive

(50, 64, 1198629)-net in base 64, using

base change [i] based on digital (34, 48, 1198629)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 8, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(50, 50+14, large)-Net over F₆₄ — Digital

Digital (50, 64, large)-net over F₆₄, using

3 times m-reduction [i] based on digital (50, 67, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁶⁷, large, F₆₄, 17) (dual of [large, large−67, 18]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(50, 50+14, large)-Net in Base 64 — Upper bound on s

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

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

Best Known (50, 50+14, s)-Nets in Base 64

(50, 50+14, 1198516)-Net over F64 — Constructive and digital

(50, 50+14, 1198629)-Net in Base 64 — Constructive

(50, 50+14, large)-Net over F64 — Digital

(50, 50+14, large)-Net in Base 64 — Upper bound on s

(50, 50+14, 1198516)-Net over F₆₄ — Constructive and digital

(50, 50+14, large)-Net over F₆₄ — Digital