MinT - Best Known (20−11, 20, s)-Nets in Base 64

Best Known (20−11, 20, s)-Nets in Base 64

(20−11, 20, 210)-Net over F₆₄ — Constructive and digital

Digital (9, 20, 210)-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, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
3. digital (1, 12, 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]

(20−11, 20, 322)-Net in Base 64 — Constructive

(9, 20, 322)-net in base 64, using

(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. (4, 15, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (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]

(20−11, 20, 425)-Net over F₆₄ — Digital

Digital (9, 20, 425)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁰, 425, F₆₄, 11) (dual of [425, 405, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(64²⁰, 585, F₆₄, 11) (dual of [585, 565, 12]-code), using
  - the BCH-code C(I) with length 585Â | 64²âˆ’1, defining interval IÂ =Â {1,8,15,…,71}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(20−11, 20, 301978)-Net in Base 64 — Upper bound on s

There is no (9, 20, 301979)-net in base 64, because

1 times m-reduction [i] would yield (9, 19, 301979)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 20769 219038 736778 615683 634524 265804 > 64¹⁹ [i]