MinT - Best Known (49−18, 49, s)-Nets in Base 16

Best Known (49−18, 49, s)-Nets in Base 16

(49−18, 49, 559)-Net over F₁₆ — Constructive and digital

Digital (31, 49, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [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]

(49−18, 49, 2049)-Net over F₁₆ — Digital

Digital (31, 49, 2049)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁹, 2049, F₁₆, 2, 18) (dual of [(2049, 2), 4049, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16⁴⁹, 4098, F₁₆, 18) (dual of [4098, 4049, 19]-code), using
  - discarding factors / shortening the dual code based on linear OA(16⁴⁹, 4099, F₁₆, 18) (dual of [4099, 4050, 19]-code), using
    - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [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(16⁴⁶, 4096, F₁₆, 17) (dual of [4096, 4050, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(49−18, 49, 994082)-Net in Base 16 — Upper bound on s

There is no (31, 49, 994083)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 100433 872301 234999 735222 135245 293048 146915 790863 917741 514056 > 16⁴⁹ [i]