MinT - Best Known (48−19, 48, s)-Nets in Base 9

Best Known (48−19, 48, s)-Nets in Base 9

(48−19, 48, 320)-Net over F₉ — Constructive and digital

Digital (29, 48, 320)-net over F₉, using

trace code for nets [i] based on digital (5, 24, 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]

(48−19, 48, 364)-Net over F₉ — Digital

Digital (29, 48, 364)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁴⁸, 364, F₉, 19) (dual of [364, 316, 20]-code), using
- the BCH-code C(I) with length 364Â | 9³âˆ’1, defining interval IÂ =Â {179,210,241,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(48−19, 48, 49875)-Net in Base 9 — Upper bound on s

There is no (29, 48, 49876)-net in base 9, because

1 times m-reduction [i] would yield (29, 47, 49876)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 707 087232 247218 547698 266688 475127 519417 552289 > 9⁴⁷ [i]