MinT - Best Known (70−14, 70, s)-Nets in Base 4

Best Known (70−14, 70, s)-Nets in Base 4

(70−14, 70, 1062)-Net over F₄ — Constructive and digital

Digital (56, 70, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 14, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 7, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
2. digital (42, 56, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 14, 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]

(70−14, 70, 5097)-Net over F₄ — Digital

Digital (56, 70, 5097)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁰, 5097, F₄, 14) (dual of [5097, 5027, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁰, 5461, F₄, 14) (dual of [5461, 5391, 15]-code), using
  - the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,40}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(70−14, 70, 1181395)-Net in Base 4 — Upper bound on s

There is no (56, 70, 1181396)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 393799 945636 395628 136788 714132 216583 468796 > 4⁷⁰ [i]