MinT - Best Known (161−136, 161, s)-Nets in Base 8

Best Known (161−136, 161, s)-Nets in Base 8

(161−136, 161, 65)-Net over F₈ — Constructive and digital

Digital (25, 161, 65)-net over F₈, using

t-expansion [i] based on digital (14, 161, 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]

(161−136, 161, 86)-Net over F₈ — Digital

Digital (25, 161, 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]

(161−136, 161, 414)-Net over F₈ — Upper bound on s (digital)

There is no digital (25, 161, 415)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8¹⁶¹, 415, F₈, 136) (dual of [415, 254, 137]-code), but
- residual code [i] would yield OA(8²⁵, 278, S₈, 17), but
  - 1 times truncation [i] would yield OA(8²⁴, 277, S₈, 16), but
    - the linear programming bound shows that MÂ â‰¥ 78638 403871 838919 782431 803777 157693 440000 / 16 504446 602386 301789 > 8²⁴ [i]

(161−136, 161, 466)-Net in Base 8 — Upper bound on s

There is no (25, 161, 467)-net in base 8, because

18 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]