MinT - Best Known (52−20, 52, s)-Nets in Base 9

Best Known (52−20, 52, s)-Nets in Base 9

(52−20, 52, 320)-Net over F₉ — Constructive and digital

Digital (32, 52, 320)-net over F₉, using

2 times m-reduction [i] based on digital (32, 54, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 27, 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]

(52−20, 52, 468)-Net over F₉ — Digital

Digital (32, 52, 468)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵², 468, F₉, 20) (dual of [468, 416, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁵², 728, F₉, 20) (dual of [728, 676, 21]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(52−20, 52, 51867)-Net in Base 9 — Upper bound on s

There is no (32, 52, 51868)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41 745846 016463 768539 522999 480181 988108 745496 074305 > 9⁵² [i]