MinT - Best Known (55−20, 55, s)-Nets in Base 9

Best Known (55−20, 55, s)-Nets in Base 9

(55−20, 55, 344)-Net over F₉ — Constructive and digital

Digital (35, 55, 344)-net over F₉, using

1 times m-reduction [i] based on digital (35, 56, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(55−20, 55, 679)-Net over F₉ — Digital

Digital (35, 55, 679)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵⁵, 679, F₉, 20) (dual of [679, 624, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁵⁵, 728, F₉, 20) (dual of [728, 673, 21]-code), using
  - the primitive BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â {−7,−6,…,12}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(55−20, 55, 100275)-Net in Base 9 — Upper bound on s

There is no (35, 55, 100276)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30433 778401 786586 568277 662561 874947 447793 523896 698561 > 9⁵⁵ [i]