MinT - Best Known (51−18, 51, s)-Nets in Base 16

Best Known (51−18, 51, s)-Nets in Base 16

(51−18, 51, 579)-Net over F₁₆ — Constructive and digital

Digital (33, 51, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 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 (18, 36, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 18, 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]

(51−18, 51, 594)-Net in Base 16 — Constructive

(33, 51, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (6, 15, 80)-net in base 16, using
  - base change [i] based on digital (1, 10, 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 (18, 36, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 18, 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]

(51−18, 51, 2618)-Net over F₁₆ — Digital

Digital (33, 51, 2618)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵¹, 2618, F₁₆, 18) (dual of [2618, 2567, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁵¹, 4104, F₁₆, 18) (dual of [4104, 4053, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(14) [i] based on
    1. 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]
    2. linear OA(16⁴³, 4096, F₁₆, 15) (dual of [4096, 4053, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(16², 8, F₁₆, 2) (dual of [8, 6, 3]-code or 8-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]

(51−18, 51, 1840796)-Net in Base 16 — Upper bound on s

There is no (33, 51, 1840797)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 711128 974392 509729 006282 590143 449628 779383 929891 035900 598396 > 16⁵¹ [i]