MinT - Best Known (67−26, 67, s)-Nets in Base 8

Best Known (67−26, 67, s)-Nets in Base 8

(67−26, 67, 354)-Net over F₈ — Constructive and digital

Digital (41, 67, 354)-net over F₈, using

1 times m-reduction [i] based on digital (41, 68, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 34, 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]

(67−26, 67, 413)-Net over F₈ — Digital

Digital (41, 67, 413)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁷, 413, F₈, 26) (dual of [413, 346, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶⁷, 511, F₈, 26) (dual of [511, 444, 27]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(67−26, 67, 36525)-Net in Base 8 — Upper bound on s

There is no (41, 67, 36526)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 213926 653958 844996 618069 509951 018142 057452 639149 182498 357963 > 8⁶⁷ [i]