MinT - Best Known (39, 39+24, s)-Nets in Base 8

Best Known (39, 39+24, s)-Nets in Base 8

(39, 39+24, 354)-Net over F₈ — Constructive and digital

Digital (39, 63, 354)-net over F₈, using

1 times m-reduction [i] based on digital (39, 64, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 32, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(39, 39+24, 442)-Net over F₈ — Digital

Digital (39, 63, 442)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶³, 442, F₈, 24) (dual of [442, 379, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶³, 511, F₈, 24) (dual of [511, 448, 25]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(39, 39+24, 41630)-Net in Base 8 — Upper bound on s

There is no (39, 63, 41631)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 774018 764018 373425 145993 938002 570702 301838 379561 654649 > 8⁶³ [i]