MinT - Best Known (5, 39, s)-Nets in Base 8

Best Known (5, 39, s)-Nets in Base 8

(5, 39, 28)-Net over F₈ — Constructive and digital

Digital (5, 39, 28)-net over F₈, using

net from sequence [i] based on digital (5, 27)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
  - a function field by SÃ©mirat [i]

(5, 39, 29)-Net over F₈ — Digital

Digital (5, 39, 29)-net over F₈, using

net from sequence [i] based on digital (5, 28)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(5, 39, 67)-Net over F₈ — Upper bound on s (digital)

There is no digital (5, 39, 68)-net over F₈, because

2 times m-reduction [i] would yield digital (5, 37, 68)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8³⁷, 68, F₈, 32) (dual of [68, 31, 33]-code), but
  - residual code [i] would yield OA(8⁵, 35, S₈, 4), but
    - the linear programming bound shows that MÂ â‰¥ 1 306624 / 39 > 8⁵ [i]

(5, 39, 82)-Net in Base 8 — Upper bound on s

There is no (5, 39, 83)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8³⁹, 83, S₈, 34), but
- the linear programming bound shows that MÂ â‰¥ 45 634012 001041 176921 080871 938056 479824 833320 287034 817933 172541 489152 / 265 829351 035282 206850 599564 523879 > 8³⁹ [i]