MinT - Best Known (196, 237, s)-Nets in Base 3

Best Known (196, 237, s)-Nets in Base 3

(196, 237, 1480)-Net over F₃ — Constructive and digital

Digital (196, 237, 1480)-net over F₃, using

3 times m-reduction [i] based on digital (196, 240, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 60, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(196, 237, 5900)-Net over F₃ — Digital

Digital (196, 237, 5900)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁷, 5900, F₃, 41) (dual of [5900, 5663, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁷, 6582, F₃, 41) (dual of [6582, 6345, 42]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3²⁰, 21, F₃, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,3)), using
      - dual of repetition code with length 21 [i]
    2. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using
      - an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]

(196, 237, 1771341)-Net in Base 3 — Upper bound on s

There is no (196, 237, 1771342)-net in base 3, because

1 times m-reduction [i] would yield (196, 236, 1771342)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 39867 529081 368939 640017 440160 641122 932114 326034 130768 314906 270243 189544 873349 284003 905565 907162 293049 833617 968553 > 3²³⁶ [i]