MinT - Best Known (75, 75+39, s)-Nets in Base 16

Best Known (75, 75+39, s)-Nets in Base 16

(75, 75+39, 612)-Net over F₁₆ — Constructive and digital

Digital (75, 114, 612)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (17, 36, 98)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 11, 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 (6, 25, 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 (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 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]

(75, 75+39, 664)-Net in Base 16 — Constructive

(75, 114, 664)-net in base 16, using

16¹ times duplication [i] based on (74, 113, 664)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (16, 35, 150)-net in base 16, using
    - base change [i] based on digital (1, 20, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (39, 78, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 39, 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]

(75, 75+39, 4238)-Net over F₁₆ — Digital

Digital (75, 114, 4238)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁴, 4238, F₁₆, 39) (dual of [4238, 4124, 40]-code), using
- 130 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 28 times 0, 1, 94 times 0) [i] based on linear OA(16¹¹⁰, 4104, F₁₆, 39) (dual of [4104, 3994, 40]-code), using
  - constructionÂ X applied to C([0,19]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(16¹⁰⁹, 4097, F₁₆, 39) (dual of [4097, 3988, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    2. linear OA(16¹⁰³, 4097, F₁₆, 37) (dual of [4097, 3994, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [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]

(75, 75+39, 7664235)-Net in Base 16 — Upper bound on s

There is no (75, 114, 7664236)-net in base 16, because

1 times m-reduction [i] would yield (75, 113, 7664236)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11629 441803 892005 107726 406542 777306 195914 349558 987080 570191 069975 403082 378257 888738 047382 307821 789023 011410 637535 250817 681890 047551 351986 > 16¹¹³ [i]