MinT - Best Known (52−19, 52, s)-Nets in Base 16

Best Known (52−19, 52, s)-Nets in Base 16

(52−19, 52, 563)-Net over F₁₆ — Constructive and digital

Digital (33, 52, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (19, 38, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 19, 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]

(52−19, 52, 579)-Net in Base 16 — Constructive

(33, 52, 579)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (5, 14, 65)-net in base 16, using
  - 1 times m-reduction [i] based on (5, 15, 65)-net in base 16, using
    - base change [i] based on digital (0, 10, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
2. digital (19, 38, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 19, 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]

(52−19, 52, 2049)-Net over F₁₆ — Digital

Digital (33, 52, 2049)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁵², 2049, F₁₆, 2, 19) (dual of [(2049, 2), 4046, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16⁵², 4098, F₁₆, 19) (dual of [4098, 4046, 20]-code), using
  - discarding factors / shortening the dual code based on linear OA(16⁵², 4099, F₁₆, 19) (dual of [4099, 4047, 20]-code), using
    - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
      1. linear OA(16⁵², 4096, F₁₆, 19) (dual of [4096, 4044, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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(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]

(52−19, 52, 1840796)-Net in Base 16 — Upper bound on s

There is no (33, 52, 1840797)-net in base 16, because

1 times m-reduction [i] would yield (33, 51, 1840797)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 711128 974392 509729 006282 590143 449628 779383 929891 035900 598396 > 16⁵¹ [i]