MinT - Best Known (12−7, 12, s)-Nets in Base 16

Best Known (12−7, 12, s)-Nets in Base 16

(12−7, 12, 51)-Net over F₁₆ — Constructive and digital

Digital (5, 12, 51)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 17)-net over F₁₆, using
  - kÂ =Â 2 construction [i]
2. digital (0, 3, 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]
3. digital (0, 7, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆ (see above)

(12−7, 12, 76)-Net over F₁₆ — Digital

Digital (5, 12, 76)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹², 76, F₁₆, 7) (dual of [76, 64, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹², 85, F₁₆, 7) (dual of [85, 73, 8]-code), using
  - the BCH-code C(I) with length 85Â | 16²âˆ’1, defining interval IÂ =Â {1,4,7,…,19}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(12−7, 12, 80)-Net in Base 16 — Constructive

(5, 12, 80)-net in base 16, using

base change [i] based on digital (1, 8, 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]

(12−7, 12, 81)-Net in Base 16

(5, 12, 81)-net in base 16, using

base change [i] based on digital (1, 8, 81)-net over F₆₄, using
- net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(12−7, 12, 3149)-Net in Base 16 — Upper bound on s

There is no (5, 12, 3150)-net in base 16, because

1 times m-reduction [i] would yield (5, 11, 3150)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 17 601527 498626 > 16¹¹ [i]