MinT - Best Known (84, 84+20, s)-Nets in Base 4

Best Known (84, 84+20, s)-Nets in Base 4

(84, 84+20, 1094)-Net over F₄ — Constructive and digital

Digital (84, 104, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (14, 24, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 12, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (60, 80, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 20, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(84, 84+20, 5460)-Net over F₄ — Digital

Digital (84, 104, 5460)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁴, 5460, F₄, 20) (dual of [5460, 5356, 21]-code), using
- 1 times truncation [i] based on linear OA(4¹⁰⁵, 5461, F₄, 21) (dual of [5461, 5356, 22]-code), using
  - the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,316,631,…,840}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(84, 84+20, 2755990)-Net in Base 4 — Upper bound on s

There is no (84, 104, 2755991)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 411 376811 960855 753709 496878 135032 056043 552479 640610 412619 694070 > 4¹⁰⁴ [i]