MinT - Best Known (47, 47+26, s)-Nets in Base 16

Best Known (47, 47+26, s)-Nets in Base 16

(47, 47+26, 581)-Net over F₁₆ — Constructive and digital

Digital (47, 73, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 19, 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 (28, 54, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 27, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(47, 47+26, 594)-Net in Base 16 — Constructive

(47, 73, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (8, 21, 80)-net in base 16, using
  - base change [i] based on digital (1, 14, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (26, 52, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 26, 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]

(47, 47+26, 2665)-Net over F₁₆ — Digital

Digital (47, 73, 2665)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷³, 2665, F₁₆, 26) (dual of [2665, 2592, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷³, 4096, F₁₆, 26) (dual of [4096, 4023, 27]-code), using
  - an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(47, 47+26, 2182279)-Net in Base 16 — Upper bound on s

There is no (47, 73, 2182280)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7957 194221 836735 466925 383833 266729 106571 327469 043303 710127 494813 920035 398084 158007 060976 > 16⁷³ [i]