MinT - Best Known (27, 42, s)-Nets in Base 9

Best Known (27, 42, s)-Nets in Base 9

(27, 42, 320)-Net over F₉ — Constructive and digital

Digital (27, 42, 320)-net over F₉, using

2 times m-reduction [i] based on digital (27, 44, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 22, 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]

(27, 42, 717)-Net over F₉ — Digital

Digital (27, 42, 717)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁴², 717, F₉, 15) (dual of [717, 675, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁴², 728, F₉, 15) (dual of [728, 686, 16]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(27, 42, 164041)-Net in Base 9 — Upper bound on s

There is no (27, 42, 164042)-net in base 9, because

1 times m-reduction [i] would yield (27, 41, 164042)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1330 321384 785349 010814 945743 886233 918705 > 9⁴¹ [i]