MinT - Best Known (14, 14+16, s)-Nets in Base 16

Best Known (14, 14+16, s)-Nets in Base 16

(14, 14+16, 82)-Net over F₁₆ — Constructive and digital

Digital (14, 30, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 22, 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]

(14, 14+16, 127)-Net over F₁₆ — Digital

Digital (14, 30, 127)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16³⁰, 127, F₁₆, 2, 16) (dual of [(127, 2), 224, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16³⁰, 254, F₁₆, 16) (dual of [254, 224, 17]-code), using
  - discarding factors / shortening the dual code based on linear OA(16³⁰, 255, F₁₆, 16) (dual of [255, 225, 17]-code), using
    - 2 times truncation [i] based on linear OA(16³², 257, F₁₆, 18) (dual of [257, 225, 19]-code), using
      - the BCH-code C(I) with length 257Â | 16⁴âˆ’1, defining interval IÂ =Â {1,16,31,…,256}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(14, 14+16, 150)-Net in Base 16 — Constructive

(14, 30, 150)-net in base 16, using

base change [i] based on (4, 20, 150)-net in base 64, using
- 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
  - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(14, 14+16, 8219)-Net in Base 16 — Upper bound on s

There is no (14, 30, 8220)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 330111 986762 277155 609232 653371 380651 > 16³⁰ [i]