MinT - Best Known (91−18, 91, s)-Nets in Base 4

Best Known (91−18, 91, s)-Nets in Base 4

(91−18, 91, 1062)-Net over F₄ — Constructive and digital

Digital (73, 91, 1062)-net over F₄, using

4¹ times duplication [i] based on digital (72, 90, 1062)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 18, 34)-net over F₄, using
    - trace code for nets [i] based on digital (0, 9, 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 (54, 72, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 18, 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]

(91−18, 91, 5461)-Net over F₄ — Digital

Digital (73, 91, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹¹, 5461, F₄, 18) (dual of [5461, 5370, 19]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,52}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(91−18, 91, 1690923)-Net in Base 4 — Upper bound on s

There is no (73, 91, 1690924)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 130010 735830 042208 822187 066609 945824 674263 682926 510469 > 4⁹¹ [i]