MinT - Best Known (141−38, 141, s)-Nets in Base 4

Best Known (141−38, 141, s)-Nets in Base 4

(141−38, 141, 531)-Net over F₄ — Constructive and digital

Digital (103, 141, 531)-net over F₄, using

3 times m-reduction [i] based on digital (103, 144, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 48, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(141−38, 141, 1018)-Net over F₄ — Digital

Digital (103, 141, 1018)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴¹, 1018, F₄, 38) (dual of [1018, 877, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁴¹, 1023, F₄, 38) (dual of [1023, 882, 39]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]

(141−38, 141, 77611)-Net in Base 4 — Upper bound on s

There is no (103, 141, 77612)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 771194 445413 550802 792805 450128 166539 403963 242625 239859 955126 640441 015526 794827 062790 > 4¹⁴¹ [i]