MinT - Best Known (36, 36+15, s)-Nets in Base 3

Best Known (36, 36+15, s)-Nets in Base 3

(36, 36+15, 144)-Net over F₃ — Constructive and digital

Digital (36, 51, 144)-net over F₃, using

trace code for nets [i] based on digital (2, 17, 48)-net over F₂₇, using
- net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
    - a function field by GarcÃa and Quoos [i]

(36, 36+15, 183)-Net over F₃ — Digital

Digital (36, 51, 183)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵¹, 183, F₃, 15) (dual of [183, 132, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵¹, 242, F₃, 15) (dual of [242, 191, 16]-code), using
  - the primitive BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â {−4,−3,…,10}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(36, 36+15, 4317)-Net in Base 3 — Upper bound on s

There is no (36, 51, 4318)-net in base 3, because

1 times m-reduction [i] would yield (36, 50, 4318)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 718336 376876 528232 460473 > 3⁵⁰ [i]