MinT - Best Known (24−9, 24, s)-Nets in Base 9

Best Known (24−9, 24, s)-Nets in Base 9

(24−9, 24, 232)-Net over F₉ — Constructive and digital

Digital (15, 24, 232)-net over F₉, using

2 times m-reduction [i] based on digital (15, 26, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 13, 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]

(24−9, 24, 574)-Net over F₉ — Digital

Digital (15, 24, 574)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9²⁴, 574, F₉, 9) (dual of [574, 550, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(9²⁴, 728, F₉, 9) (dual of [728, 704, 10]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(24−9, 24, 84888)-Net in Base 9 — Upper bound on s

There is no (15, 24, 84889)-net in base 9, because

1 times m-reduction [i] would yield (15, 23, 84889)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8863 297034 042616 468129 > 9²³ [i]