MinT - Best Known (16, 16+11, s)-Nets in Base 9

Best Known (16, 16+11, s)-Nets in Base 9

(16, 16+11, 232)-Net over F₉ — Constructive and digital

Digital (16, 27, 232)-net over F₉, using

1 times m-reduction [i] based on digital (16, 28, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 14, 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]

(16, 16+11, 292)-Net over F₉ — Digital

Digital (16, 27, 292)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9²⁷, 292, F₉, 11) (dual of [292, 265, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(9²⁷, 364, F₉, 11) (dual of [364, 337, 12]-code), using
  - the BCH-code C(I) with length 364Â | 9³âˆ’1, defining interval IÂ =Â {171,264,357,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(16, 16+11, 29837)-Net in Base 9 — Upper bound on s

There is no (16, 27, 29838)-net in base 9, because

1 times m-reduction [i] would yield (16, 26, 29838)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 461126 710277 848328 690865 > 9²⁶ [i]