MinT - Best Known (1, ∞, s)-Nets in Base 16

Best Known (1, ∞, s)-Nets in Base 16

(1, ∞, 24)-Net over F₁₆ — Constructive and digital

Digital (1, m, 24)-net over F₁₆ for arbitrarily large m, 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]

(1, ∞, 25)-Net over F₁₆ — Digital

Digital (1, m, 25)-net over F₁₆ for arbitrarily large m, using

net from sequence [i] based on digital (1, 24)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 25, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(1, ∞, 33)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (1, m, 34)-net over F₁₆ for arbitrarily large m, because

m-reduction [i] would yield digital (1, 31, 34)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16³¹, 34, F₁₆, 30) (dual of [34, 3, 31]-code), but
  - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(1, ∞, 34)-Net in Base 16 — Upper bound on s

There is no (1, m, 35)-net in base 16 for arbitrarily large m, because

m-reduction [i] would yield (1, 33, 35)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16³³, 35, S₁₆, 32), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 87112 285931 760246 646623 899502 532662 132736 / 11 > 16³³ [i]