MinT - Best Known (77, 77+28, s)-Nets in Base 4

Best Known (77, 77+28, s)-Nets in Base 4

(77, 77+28, 531)-Net over F₄ — Constructive and digital

Digital (77, 105, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 35, 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]

(77, 77+28, 881)-Net over F₄ — Digital

Digital (77, 105, 881)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁵, 881, F₄, 28) (dual of [881, 776, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰⁵, 1023, F₄, 28) (dual of [1023, 918, 29]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(77, 77+28, 66025)-Net in Base 4 — Upper bound on s

There is no (77, 105, 66026)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 611022 661624 040121 719096 515404 461564 110079 263562 318558 141616 > 4¹⁰⁵ [i]