MinT - Best Known (34, 34+21, s)-Nets in Base 9

Best Known (34, 34+21, s)-Nets in Base 9

(34, 34+21, 320)-Net over F₉ — Constructive and digital

Digital (34, 55, 320)-net over F₉, using

3 times m-reduction [i] based on digital (34, 58, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 29, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(34, 34+21, 501)-Net over F₉ — Digital

Digital (34, 55, 501)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵⁵, 501, F₉, 21) (dual of [501, 446, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁵⁵, 728, F₉, 21) (dual of [728, 673, 22]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(34, 34+21, 80494)-Net in Base 9 — Upper bound on s

There is no (34, 55, 80495)-net in base 9, because

1 times m-reduction [i] would yield (34, 54, 80495)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3381 707928 734121 913152 275340 585187 097845 463509 183665 > 9⁵⁴ [i]