MinT - Best Known (27−15, 27, s)-Nets in Base 64

Best Known (27−15, 27, s)-Nets in Base 64

(27−15, 27, 195)-Net over F₆₄ — Constructive and digital

Digital (12, 27, 195)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 5, 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, 7, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
3. digital (0, 15, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(27−15, 27, 322)-Net in Base 64 — Constructive

(12, 27, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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. (5, 20, 257)-net in base 64, using
  - base change [i] based on digital (0, 15, 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]

(27−15, 27, 363)-Net over F₆₄ — Digital

Digital (12, 27, 363)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁷, 363, F₆₄, 15) (dual of [363, 336, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(64²⁷, 455, F₆₄, 15) (dual of [455, 428, 16]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,38,75,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(27−15, 27, 274310)-Net in Base 64 — Upper bound on s

There is no (12, 27, 274311)-net in base 64, because

1 times m-reduction [i] would yield (12, 26, 274311)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 91346 171937 723768 527106 201172 095312 308710 139200 > 64²⁶ [i]