MinT - Best Known (40−22, 40, s)-Nets in Base 16

Best Known (40−22, 40, s)-Nets in Base 16

(40−22, 40, 89)-Net over F₁₆ — Constructive and digital

Digital (18, 40, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 28, 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]

(40−22, 40, 126)-Net over F₁₆ — Digital

Digital (18, 40, 126)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁰, 126, F₁₆, 2, 22) (dual of [(126, 2), 212, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16⁴⁰, 129, F₁₆, 2, 22) (dual of [(129, 2), 218, 23]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(16⁴⁰, 258, F₁₆, 22) (dual of [258, 218, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
      1. linear OA(16⁴⁰, 256, F₁₆, 22) (dual of [256, 216, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 255Â = 16²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(16³⁸, 256, F₁₆, 21) (dual of [256, 218, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 255Â = 16²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(16⁰, 2, F₁₆, 0) (dual of [2, 2, 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]

(40−22, 40, 129)-Net in Base 16 — Constructive

(18, 40, 129)-net in base 16, using

2 times m-reduction [i] based on (18, 42, 129)-net in base 16, using
- base change [i] based on digital (0, 24, 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]

(40−22, 40, 7820)-Net in Base 16 — Upper bound on s

There is no (18, 40, 7821)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 462963 008541 770842 277240 200465 160203 692008 149716 > 16⁴⁰ [i]