MinT - Best Known (22, 22+29, s)-Nets in Base 64

Best Known (22, 22+29, s)-Nets in Base 64

Digital (22, 51, 257)-net over F₆₄, using

(22, 51, 288)-net in base 64, using

40 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

Digital (22, 51, 366)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵¹, 366, F₆₄, 29) (dual of [366, 315, 30]-code), using
- 22 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 21 times 0) [i] based on linear OA(64⁴⁹, 342, F₆₄, 29) (dual of [342, 293, 30]-code), using
  - extended algebraic-geometric code AG_e(F,312P) [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(22, 51, 513)-net in base 64, using

5 times m-reduction [i] based on (22, 56, 513)-net in base 64, using
- base change [i] based on digital (8, 42, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

There is no (22, 51, 270863)-net in base 64, because

1 times m-reduction [i] would yield (22, 50, 270863)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 037069 342168 699877 370185 148225 464503 837028 615292 184216 920532 798726 601868 922994 489344 464833 > 64⁵⁰ [i]