MinT - Best Known (6, 14, s)-Nets in Base 64

Best Known (6, 14, s)-Nets in Base 64

(6, 14, 195)-Net over F₆₄ — Constructive and digital

Digital (6, 14, 195)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
2. 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]
3. digital (0, 8, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(6, 14, 259)-Net in Base 64 — Constructive

(6, 14, 259)-net in base 64, using

2 times m-reduction [i] based on (6, 16, 259)-net in base 64, using
- base change [i] based on digital (2, 12, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(6, 14, 387)-Net over F₆₄ — Digital

Digital (6, 14, 387)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁴, 387, F₆₄, 8) (dual of [387, 373, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹⁴, 455, F₆₄, 8) (dual of [455, 441, 9]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,10,19,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(6, 14, 73677)-Net in Base 64 — Upper bound on s

There is no (6, 14, 73678)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 19 343681 505676 419427 422811 > 64¹⁴ [i]