MinT - Best Known (10, 22, s)-Nets in Base 64

Best Known (10, 22, s)-Nets in Base 64

(10, 22, 195)-Net over F₆₄ — Constructive and digital

Digital (10, 22, 195)-net over F₆₄, using

1 times m-reduction [i] based on digital (10, 23, 195)-net over F₆₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 4, 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 (0, 6, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
  3. digital (0, 13, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(10, 22, 322)-Net in Base 64 — Constructive

(10, 22, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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. (4, 16, 257)-net in base 64, using
  - base change [i] based on digital (0, 12, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(10, 22, 442)-Net over F₆₄ — Digital

Digital (10, 22, 442)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²², 442, F₆₄, 12) (dual of [442, 420, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(64²², 585, F₆₄, 12) (dual of [585, 563, 13]-code), using
  - the BCH-code C(I) with length 585Â | 64²âˆ’1, defining interval IÂ =Â {1,8,15,…,78}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(10, 22, 199313)-Net in Base 64 — Upper bound on s

There is no (10, 22, 199314)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 5444 669961 864660 747052 208719 070316 726040 > 64²² [i]