MinT - Best Known (40, 40+23, s)-Nets in Base 9

Best Known (40, 40+23, s)-Nets in Base 9

(40, 40+23, 344)-Net over F₉ — Constructive and digital

Digital (40, 63, 344)-net over F₉, using

3 times m-reduction [i] based on digital (40, 66, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 33, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(40, 40+23, 701)-Net over F₉ — Digital

Digital (40, 63, 701)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶³, 701, F₉, 23) (dual of [701, 638, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁶³, 728, F₉, 23) (dual of [728, 665, 24]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(40, 40+23, 146677)-Net in Base 9 — Upper bound on s

There is no (40, 63, 146678)-net in base 9, because

1 times m-reduction [i] would yield (40, 62, 146678)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 145564 856982 664860 676039 080849 102166 322948 489763 270013 603665 > 9⁶² [i]