MinT - Best Known (76, 76+40, s)-Nets in Base 16

Best Known (76, 76+40, s)-Nets in Base 16

(76, 76+40, 596)-Net over F₁₆ — Constructive and digital

Digital (76, 116, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (16, 36, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 10, 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 (6, 26, 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 (40, 80, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 40, 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]

(76, 76+40, 643)-Net in Base 16 — Constructive

(76, 116, 643)-net in base 16, using

1 times m-reduction [i] based on (76, 117, 643)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (15, 35, 129)-net in base 16, using
    - base change [i] based on digital (0, 20, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. digital (41, 82, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 41, 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]

(76, 76+40, 4161)-Net over F₁₆ — Digital

Digital (76, 116, 4161)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁶, 4161, F₁₆, 40) (dual of [4161, 4045, 41]-code), using
- 58 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 10 times 0, 1, 43 times 0) [i] based on linear OA(16¹¹², 4099, F₁₆, 40) (dual of [4099, 3987, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(38) [i] based on
    1. linear OA(16¹¹², 4096, F₁₆, 40) (dual of [4096, 3984, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(16¹⁰⁹, 4096, F₁₆, 39) (dual of [4096, 3987, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(76, 76+40, 5334699)-Net in Base 16 — Upper bound on s

There is no (76, 116, 5334700)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 634129 796249 379030 520188 235163 812845 922422 629763 425000 639916 342272 118149 902354 488855 056266 753605 148105 368694 778585 010472 138889 011999 106876 > 16¹¹⁶ [i]