MinT - Best Known (30, 30+17, s)-Nets in Base 64

Best Known (30, 30+17, s)-Nets in Base 64

(30, 30+17, 780)-Net over F₆₄ — Constructive and digital

Digital (30, 47, 780)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 65)-net over F₆₄, using
  - s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 1, 65)-net over F₆₄ (see above)
3. digital (0, 1, 65)-net over F₆₄ (see above)
4. digital (0, 1, 65)-net over F₆₄ (see above)
5. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
6. digital (0, 2, 65)-net over F₆₄ (see above)
7. digital (0, 2, 65)-net over F₆₄ (see above)
8. 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]
9. digital (0, 4, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
10. digital (0, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
11. digital (0, 8, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
12. digital (0, 17, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(30, 30+17, 8192)-Net in Base 64 — Constructive

(30, 47, 8192)-net in base 64, using

64³ times duplication [i] based on (27, 44, 8192)-net in base 64, using
- base change [i] based on digital (16, 33, 8192)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³³, 8192, F₂₅₆, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(256³³, 65537, F₂₅₆, 17) (dual of [65537, 65504, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(30, 30+17, 21828)-Net over F₆₄ — Digital

Digital (30, 47, 21828)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 30+17, large)-Net in Base 64 — Upper bound on s

There is no (30, 47, large)-net in base 64, because

15 times m-reduction [i] would yield (30, 32, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]