MinT - Best Known (7, 7+9, s)-Nets in Base 64

Best Known (7, 7+9, s)-Nets in Base 64

(7, 7+9, 195)-Net over F₆₄ — Constructive and digital

Digital (7, 16, 195)-net over F₆₄, using

generalized (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 (0, 4, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
3. digital (0, 9, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(7, 7+9, 322)-Net in Base 64 — Constructive

(7, 16, 322)-net in base 64, using

(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. (3, 12, 257)-net in base 64, using
  - base change [i] based on digital (0, 9, 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]

(7, 7+9, 396)-Net over F₆₄ — Digital

Digital (7, 16, 396)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁶, 396, F₆₄, 9) (dual of [396, 380, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹⁶, 455, F₆₄, 9) (dual of [455, 439, 10]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,10,19,…,73}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(7, 7+9, 208393)-Net in Base 64 — Upper bound on s

There is no (7, 16, 208394)-net in base 64, because

1 times m-reduction [i] would yield (7, 15, 208394)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1237 954926 337249 774049 331454 > 64¹⁵ [i]