MinT - Best Known (141−31, 141, s)-Nets in Base 3

Best Known (141−31, 141, s)-Nets in Base 3

(141−31, 141, 600)-Net over F₃ — Constructive and digital

Digital (110, 141, 600)-net over F₃, using

3¹ times duplication [i] based on digital (109, 140, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 35, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(141−31, 141, 1147)-Net over F₃ — Digital

Digital (110, 141, 1147)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴¹, 1147, F₃, 31) (dual of [1147, 1006, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴¹, 2187, F₃, 31) (dual of [2187, 2046, 32]-code), using
  - an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(141−31, 141, 91158)-Net in Base 3 — Upper bound on s

There is no (110, 141, 91159)-net in base 3, because

1 times m-reduction [i] would yield (110, 140, 91159)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 265845 171802 550342 917512 479503 903558 090980 608024 586196 121947 080771 > 3¹⁴⁰ [i]