MinT - Best Known (8, 19, s)-Nets in Base 16

Best Known (8, 19, s)-Nets in Base 16

(8, 19, 65)-Net over F₁₆ — Constructive and digital

Digital (8, 19, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 19, 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]

(8, 19, 68)-Net over F₁₆ — Digital

Digital (8, 19, 68)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁹, 68, F₁₆, 11) (dual of [68, 49, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹⁹, 85, F₁₆, 11) (dual of [85, 66, 12]-code), using
  - the BCH-code C(I) with length 85Â | 16²âˆ’1, defining interval IÂ =Â {16,23,30,…,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(8, 19, 80)-Net in Base 16 — Constructive

(8, 19, 80)-net in base 16, using

2 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
- base change [i] based on digital (1, 14, 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]

(8, 19, 81)-Net in Base 16

(8, 19, 81)-net in base 16, using

2 times m-reduction [i] based on (8, 21, 81)-net in base 16, using
- base change [i] based on digital (1, 14, 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]

(8, 19, 3752)-Net in Base 16 — Upper bound on s

There is no (8, 19, 3753)-net in base 16, because

1 times m-reduction [i] would yield (8, 18, 3753)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4726 256054 195595 219976 > 16¹⁸ [i]