MinT - Best Known (12, 26, s)-Nets in Base 64

Best Known (12, 26, s)-Nets in Base 64

(12, 26, 210)-Net over F₆₄ — Constructive and digital

Digital (12, 26, 210)-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, 7, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
3. digital (1, 15, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(12, 26, 322)-Net in Base 64 — Constructive

(12, 26, 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, 19, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (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]

(12, 26, 481)-Net over F₆₄ — Digital

Digital (12, 26, 481)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁶, 481, F₆₄, 14) (dual of [481, 455, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(64²⁶, 819, F₆₄, 14) (dual of [819, 793, 15]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(12, 26, 274310)-Net in Base 64 — Upper bound on s

There is no (12, 26, 274311)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 91346 171937 723768 527106 201172 095312 308710 139200 > 64²⁶ [i]