MinT - Best Known (25, 173, s)-Nets in Base 8

Best Known (25, 173, s)-Nets in Base 8

(25, 173, 65)-Net over F₈ — Constructive and digital

Digital (25, 173, 65)-net over F₈, using

t-expansion [i] based on digital (14, 173, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(25, 173, 86)-Net over F₈ — Digital

Digital (25, 173, 86)-net over F₈, using

net from sequence [i] based on digital (25, 85)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 25 and N(F) ≥ 86, using
  - a curve from the manYPoints database [i]

(25, 173, 338)-Net over F₈ — Upper bound on s (digital)

There is no digital (25, 173, 339)-net over F₈, because

4 times m-reduction [i] would yield digital (25, 169, 339)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁶⁹, 339, F₈, 144) (dual of [339, 170, 145]-code), but
  - residual code [i] would yield OA(8²⁵, 194, S₈, 18), but
    - the linear programming bound shows that MÂ â‰¥ 981 557937 430649 389126 891450 072096 402915 196928 / 25377 967658 307417 319547 > 8²⁵ [i]

(25, 173, 466)-Net in Base 8 — Upper bound on s

There is no (25, 173, 467)-net in base 8, because

30 times m-reduction [i] would yield (25, 143, 467)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1387 994308 584460 028106 920653 117892 514369 247959 049737 779053 772907 081132 280836 298479 079100 816384 514316 382881 595992 537248 121101 435776 > 8¹⁴³ [i]