MinT - Best Known (29, 46, s)-Nets in Base 9

Best Known (29, 46, s)-Nets in Base 9

(29, 46, 320)-Net over F₉ — Constructive and digital

Digital (29, 46, 320)-net over F₉, using

2 times m-reduction [i] based on digital (29, 48, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 24, 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]

(29, 46, 577)-Net over F₉ — Digital

Digital (29, 46, 577)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁴⁶, 577, F₉, 17) (dual of [577, 531, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁴⁶, 728, F₉, 17) (dual of [728, 682, 18]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(29, 46, 109697)-Net in Base 9 — Upper bound on s

There is no (29, 46, 109698)-net in base 9, because

1 times m-reduction [i] would yield (29, 45, 109698)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 728338 264429 804665 339053 807607 263323 592065 > 9⁴⁵ [i]