MinT - Best Known (50, 50+12, s)-Nets in Base 4

Best Known (50, 50+12, s)-Nets in Base 4

(50, 50+12, 1076)-Net over F₄ — Constructive and digital

Digital (50, 62, 1076)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 7, 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 (36, 48, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 12, 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]

(50, 50+12, 5460)-Net over F₄ — Digital

Digital (50, 62, 5460)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶², 5460, F₄, 12) (dual of [5460, 5398, 13]-code), using
- 1 times truncation [i] based on linear OA(4⁶³, 5461, F₄, 13) (dual of [5461, 5398, 14]-code), using
  - the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,586,1171,…,1560}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(50, 50+12, 1661063)-Net in Base 4 — Upper bound on s

There is no (50, 62, 1661064)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21 267688 305367 711243 586915 089895 687815 > 4⁶² [i]