MinT - Best Known (167, 202, s)-Nets in Base 3

Best Known (167, 202, s)-Nets in Base 3

(167, 202, 896)-Net over F₃ — Constructive and digital

Digital (167, 202, 896)-net over F₃, using

t-expansion [i] based on digital (166, 202, 896)-net over F₃, using
- 2 times m-reduction [i] based on digital (166, 204, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 51, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(167, 202, 5270)-Net over F₃ — Digital

Digital (167, 202, 5270)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰², 5270, F₃, 35) (dual of [5270, 5068, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰², 6579, F₃, 35) (dual of [6579, 6377, 36]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁷, 18, F₃, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,3)), using
      - dual of repetition code with length 18 [i]
    2. linear OA(3¹⁸⁵, 6561, F₃, 35) (dual of [6561, 6376, 36]-code), using
      - an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(167, 202, 1570969)-Net in Base 3 — Upper bound on s

There is no (167, 202, 1570970)-net in base 3, because

1 times m-reduction [i] would yield (167, 201, 1570970)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 796843 642726 787131 155973 548079 093858 748585 624173 851777 970493 852876 694589 517176 670402 031362 327445 > 3²⁰¹ [i]