MinT - Best Known (119−23, 119, s)-Nets in Base 4

Best Known (119−23, 119, s)-Nets in Base 4

(119−23, 119, 1094)-Net over F₄ — Constructive and digital

Digital (96, 119, 1094)-net over F₄, using

4¹ times duplication [i] based on digital (95, 118, 1094)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 26, 66)-net over F₄, using
    - trace code for nets [i] based on digital (2, 13, 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 (69, 92, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 23, 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]

(119−23, 119, 5461)-Net over F₄ — Digital

Digital (96, 119, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁹, 5461, F₄, 23) (dual of [5461, 5342, 24]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,67}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(119−23, 119, 4702770)-Net in Base 4 — Upper bound on s

There is no (96, 119, 4702771)-net in base 4, because

1 times m-reduction [i] would yield (96, 118, 4702771)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 110427 955973 304710 237982 692740 549019 816885 385216 493715 880971 931678 085384 > 4¹¹⁸ [i]