MinT - Best Known (112−22, 112, s)-Nets in Base 4

Best Known (112−22, 112, s)-Nets in Base 4

(112−22, 112, 1076)-Net over F₄ — Constructive and digital

Digital (90, 112, 1076)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 24, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 12, 24)-net over F₁₆, using
    - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
2. digital (66, 88, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 22, 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]

(112−22, 112, 5461)-Net over F₄ — Digital

Digital (90, 112, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹², 5461, F₄, 22) (dual of [5461, 5349, 23]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(112−22, 112, 2207783)-Net in Base 4 — Upper bound on s

There is no (90, 112, 2207784)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 26 959959 473796 102535 022116 214993 098756 766501 745641 857846 906685 313940 > 4¹¹² [i]