MinT - Best Known (81, 81+44, s)-Nets in Base 16

Best Known (81, 81+44, s)-Nets in Base 16

(81, 81+44, 587)-Net over F₁₆ — Constructive and digital

Digital (81, 125, 587)-net over F₁₆, using

1 times m-reduction [i] based on digital (81, 126, 587)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (53, 98, 522)-net over F₁₆, using
    - trace code for nets [i] based on digital (4, 49, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(81, 81+44, 612)-Net in Base 16 — Constructive

(81, 125, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 37, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (15, 40, 98)-net in base 16, using
    - base change [i] based on digital (7, 32, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
2. digital (44, 88, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 44, 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]

(81, 81+44, 3929)-Net over F₁₆ — Digital

Digital (81, 125, 3929)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁵, 3929, F₁₆, 44) (dual of [3929, 3804, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹²⁵, 4103, F₁₆, 44) (dual of [4103, 3978, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(41) [i] based on
    1. linear OA(16¹²⁴, 4096, F₁₆, 44) (dual of [4096, 3972, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(16¹¹⁸, 4096, F₁₆, 42) (dual of [4096, 3978, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(81, 81+44, 4191325)-Net in Base 16 — Upper bound on s

There is no (81, 125, 4191326)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 273407 682148 109817 994067 419339 078194 246460 219882 475863 276180 710388 089718 387889 980986 658723 872178 097628 919705 661455 068602 337642 837099 154916 927307 054556 > 16¹²⁵ [i]