MinT - Best Known (58−57, 58, s)-Nets in Base 16

Best Known (58−57, 58, s)-Nets in Base 16

(58−57, 58, 24)-Net over F₁₆ — Constructive and digital

Digital (1, 58, 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]

(58−57, 58, 25)-Net over F₁₆ — Digital

Digital (1, 58, 25)-net over F₁₆, 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]

(58−57, 58, 33)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (1, 58, 34)-net over F₁₆, because

27 times 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]

(58−57, 58, 34)-Net in Base 16 — Upper bound on s

There is no (1, 58, 35)-net in base 16, because

25 times 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]