MinT - Best Known (48, 48+25, s)-Nets in Base 16

Best Known (48, 48+25, s)-Nets in Base 16

(48, 48+25, 771)-Net over F₁₆ — Constructive and digital

Digital (48, 73, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (11, 23, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(11,256) in PG(22,16)) 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]
2. digital (25, 50, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(48, 48+25, 3685)-Net over F₁₆ — Digital

Digital (48, 73, 3685)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷³, 3685, F₁₆, 25) (dual of [3685, 3612, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷³, 4097, F₁₆, 25) (dual of [4097, 4024, 26]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(48, 48+25, 5915474)-Net in Base 16 — Upper bound on s

There is no (48, 73, 5915475)-net in base 16, because

1 times m-reduction [i] would yield (48, 72, 5915475)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 497 323720 855806 837620 702487 731121 447792 961244 003720 514732 911951 927366 639875 007138 749876 > 16⁷² [i]