MinT - Best Known (21, 21+11, s)-Nets in Base 3

Best Known (21, 21+11, s)-Nets in Base 3

(21, 21+11, 64)-Net over F₃ — Constructive and digital

Digital (21, 32, 64)-net over F₃, using

trace code for nets [i] based on digital (5, 16, 32)-net over F₉, using
- net from sequence [i] based on digital (5, 31)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
    - a function field by GarcÃa and Quoos [i]

(21, 21+11, 85)-Net over F₃ — Digital

Digital (21, 32, 85)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³², 85, F₃, 11) (dual of [85, 53, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3³², 87, F₃, 11) (dual of [87, 55, 12]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3⁵, 6, F₃, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
      - dual of repetition code with length 6 [i]
    2. linear OA(3²⁷, 81, F₃, 11) (dual of [81, 54, 12]-code), using
      - an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(21, 21+11, 1178)-Net in Base 3 — Upper bound on s

There is no (21, 32, 1179)-net in base 3, because

1 times m-reduction [i] would yield (21, 31, 1179)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 619 151437 355503 > 3³¹ [i]