MinT - Best Known (17−15, 17, s)-Nets in Base 8

Best Known (17−15, 17, s)-Nets in Base 8

(17−15, 17, 17)-Net over F₈ — Constructive and digital

Digital (2, 17, 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]

(17−15, 17, 18)-Net over F₈ — Digital

Digital (2, 17, 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]

(17−15, 17, 31)-Net over F₈ — Upper bound on s (digital)

There is no digital (2, 17, 32)-net over F₈, because

3 times m-reduction [i] would yield digital (2, 14, 32)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁴, 32, F₈, 12) (dual of [32, 18, 13]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8¹³, 16, F₈, 12) (dual of [16, 3, 13]-code), but
      - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(8¹⁸, 32, F₈, 16) (dual of [32, 14, 17]-code), but
      - discarding factors / shortening the dual code 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]

(17−15, 17, 51)-Net in Base 8 — Upper bound on s

There is no (2, 17, 52)-net in base 8, because

3 times m-reduction [i] would yield (2, 14, 52)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 666928 710900 > 8¹⁴ [i]