MinT - Best Known (25, 39, s)-Nets in Base 9

Best Known (25, 39, s)-Nets in Base 9

(25, 39, 320)-Net over F₉ — Constructive and digital

Digital (25, 39, 320)-net over F₉, using

1 times m-reduction [i] based on digital (25, 40, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 20, 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]

(25, 39, 689)-Net over F₉ — Digital

Digital (25, 39, 689)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³⁹, 689, F₉, 14) (dual of [689, 650, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(9³⁹, 728, F₉, 14) (dual of [728, 689, 15]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(25, 39, 87559)-Net in Base 9 — Upper bound on s

There is no (25, 39, 87560)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 16 423820 007847 680221 698349 785258 131137 > 9³⁹ [i]