MinT - Best Known (39, 39+21, s)-Nets in Base 16

Best Known (39, 39+21, s)-Nets in Base 16

(39, 39+21, 581)-Net over F₁₆ — Constructive and digital

Digital (39, 60, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 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 (23, 44, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 22, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(39, 39+21, 643)-Net in Base 16 — Constructive

(39, 60, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (8, 18, 129)-net in base 16, using
  - base change [i] based on (2, 12, 129)-net in base 64, using
    - 2 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
      - base change [i] based on digital (0, 12, 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 (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(39, 39+21, 2889)-Net over F₁₆ — Digital

Digital (39, 60, 2889)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁰, 2889, F₁₆, 21) (dual of [2889, 2829, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁶⁰, 4107, F₁₆, 21) (dual of [4107, 4047, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
    1. linear OA(16⁵⁸, 4096, F₁₆, 21) (dual of [4096, 4038, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(16⁴⁹, 4096, F₁₆, 18) (dual of [4096, 4047, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(16², 11, F₁₆, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
      - discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]

(39, 39+21, 3838772)-Net in Base 16 — Upper bound on s

There is no (39, 60, 3838773)-net in base 16, because

1 times m-reduction [i] would yield (39, 59, 3838773)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110428 130101 916509 150153 426328 155835 369309 455494 123523 045607 109277 626076 > 16⁵⁹ [i]