MinT - Best Known (21, 21+13, s)-Nets in Base 9

Best Known (21, 21+13, s)-Nets in Base 9

(21, 21+13, 300)-Net over F₉ — Constructive and digital

Digital (21, 34, 300)-net over F₉, using

trace code for nets [i] based on digital (4, 17, 150)-net over F₈₁, using
- net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(21, 21+13, 442)-Net over F₉ — Digital

Digital (21, 34, 442)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³⁴, 442, F₉, 13) (dual of [442, 408, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(9³⁴, 728, F₉, 13) (dual of [728, 694, 14]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(21, 21+13, 66289)-Net in Base 9 — Upper bound on s

There is no (21, 34, 66290)-net in base 9, because

1 times m-reduction [i] would yield (21, 33, 66290)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30 904611 417895 903727 553906 840225 > 9³³ [i]