MinT - Best Known (64−25, 64, s)-Nets in Base 8

Best Known (64−25, 64, s)-Nets in Base 8

(64−25, 64, 354)-Net over F₈ — Constructive and digital

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]

(64−25, 64, 389)-Net over F₈ — Digital

Digital (39, 64, 389)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁴, 389, F₈, 25) (dual of [389, 325, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶⁴, 511, F₈, 25) (dual of [511, 447, 26]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(64−25, 64, 41630)-Net in Base 8 — Upper bound on s

There is no (39, 64, 41631)-net in base 8, because

1 times m-reduction [i] would yield (39, 63, 41631)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 774018 764018 373425 145993 938002 570702 301838 379561 654649 > 8⁶³ [i]