MinT - Best Known (17, 28, s)-Nets in Base 9

Best Known (17, 28, s)-Nets in Base 9

(17, 28, 232)-Net over F₉ — Constructive and digital

Digital (17, 28, 232)-net over F₉, using

2 times m-reduction [i] based on digital (17, 30, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 15, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(17, 28, 373)-Net over F₉ — Digital

Digital (17, 28, 373)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9²⁸, 373, F₉, 11) (dual of [373, 345, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(9²⁸, 728, F₉, 11) (dual of [728, 700, 12]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(17, 28, 46305)-Net in Base 9 — Upper bound on s

There is no (17, 28, 46306)-net in base 9, because

1 times m-reduction [i] would yield (17, 27, 46306)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 58 153802 453854 533620 966097 > 9²⁷ [i]