MinT - Best Known (33, 33+19, s)-Nets in Base 9

Best Known (33, 33+19, s)-Nets in Base 9

(33, 33+19, 344)-Net over F₉ — Constructive and digital

Digital (33, 52, 344)-net over F₉, using

trace code for nets [i] based on digital (7, 26, 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]

(33, 33+19, 645)-Net over F₉ — Digital

Digital (33, 52, 645)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵², 645, F₉, 19) (dual of [645, 593, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁵², 728, F₉, 19) (dual of [728, 676, 20]-code), using
  - the primitive BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â {−6,−5,…,12}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(33, 33+19, 132439)-Net in Base 9 — Upper bound on s

There is no (33, 52, 132440)-net in base 9, because

1 times m-reduction [i] would yield (33, 51, 132440)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 638426 146537 560880 812026 283706 766494 856320 003265 > 9⁵¹ [i]