MinT - Best Known (21−19, 21, s)-Nets in Base 8

Best Known (21−19, 21, s)-Nets in Base 8

(21−19, 21, 17)-Net over F₈ — Constructive and digital

Digital (2, 21, 17)-net over F₈, using

net from sequence [i] based on digital (2, 16)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
  - a function field by SÃ©mirat [i]

(21−19, 21, 18)-Net over F₈ — Digital

Digital (2, 21, 18)-net over F₈, using

net from sequence [i] based on digital (2, 17)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 18, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(21−19, 21, 26)-Net over F₈ — Upper bound on s (digital)

There is no digital (2, 21, 27)-net over F₈, because

3 times m-reduction [i] would yield digital (2, 18, 27)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁸, 27, F₈, 16) (dual of [27, 9, 17]-code), but
  - residual code [i] would yield OA(8², 10, S₈, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 71 > 8² [i]

(21−19, 21, 45)-Net in Base 8 — Upper bound on s

There is no (2, 21, 46)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8²¹, 46, S₈, 19), but
- the linear programming bound shows that MÂ â‰¥ 14529 091252 314943 503823 011840 / 1479 210359 > 8²¹ [i]