MinT - Best Known (89, 121, s)-Nets in Base 4

Best Known (89, 121, s)-Nets in Base 4

(89, 121, 531)-Net over F₄ — Constructive and digital

Digital (89, 121, 531)-net over F₄, using

2 times m-reduction [i] based on digital (89, 123, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 41, 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]

(89, 121, 1005)-Net over F₄ — Digital

Digital (89, 121, 1005)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²¹, 1005, F₄, 32) (dual of [1005, 884, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²¹, 1023, F₄, 32) (dual of [1023, 902, 33]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {311,312,…,342}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(89, 121, 80989)-Net in Base 4 — Upper bound on s

There is no (89, 121, 80990)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 068443 220599 892192 244021 249513 213904 947773 986947 600193 104067 353391 049535 > 4¹²¹ [i]