MinT - Best Known (55−12, 55, s)-Nets in Base 64

Best Known (55−12, 55, s)-Nets in Base 64

(55−12, 55, 1398245)-Net over F₆₄ — Constructive and digital

Digital (43, 55, 1398245)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 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, 7, 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 (33, 45, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(55−12, 55, 1398358)-Net in Base 64 — Constructive

(43, 55, 1398358)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (4, 10, 258)-net in base 64, using
  - 2 times m-reduction [i] based on (4, 12, 258)-net in base 64, using
    - base change [i] based on digital (1, 9, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (33, 45, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(55−12, 55, large)-Net over F₆₄ — Digital

Digital (43, 55, large)-net over F₆₄, using

t-expansion [i] based on digital (41, 55, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁵, large, F₆₄, 14) (dual of [large, large−55, 15]-code), using
  - 2 times code embedding in larger space [i] 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]

(55−12, 55, large)-Net in Base 64 — Upper bound on s

There is no (43, 55, large)-net in base 64, because

10 times m-reduction [i] would yield (43, 45, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (55−12, 55, s)-Nets in Base 64

(55−12, 55, 1398245)-Net over F64 — Constructive and digital

(55−12, 55, 1398358)-Net in Base 64 — Constructive

(55−12, 55, large)-Net over F64 — Digital

(55−12, 55, large)-Net in Base 64 — Upper bound on s

(55−12, 55, 1398245)-Net over F₆₄ — Constructive and digital

(55−12, 55, large)-Net over F₆₄ — Digital