MinT - Best Known (216−34, 216, s)-Nets in Base 3

Best Known (216−34, 216, s)-Nets in Base 3

(216−34, 216, 1480)-Net over F₃ — Constructive and digital

Digital (182, 216, 1480)-net over F₃, using

t-expansion [i] based on digital (181, 216, 1480)-net over F₃, using
- 4 times m-reduction [i] based on digital (181, 220, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 55, 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]

(216−34, 216, 10238)-Net over F₃ — Digital

Digital (182, 216, 10238)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁶, 10238, F₃, 34) (dual of [10238, 10022, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁶, 19701, F₃, 34) (dual of [19701, 19485, 35]-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(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using
      - an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(216−34, 216, 4141522)-Net in Base 3 — Upper bound on s

There is no (182, 216, 4141523)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 433840 873338 067011 128140 117223 751249 536112 620995 900657 518109 138858 090169 634997 363533 880924 758437 568007 > 3²¹⁶ [i]