MinT - Best Known (21, 21+128, s)-Nets in Base 9

Best Known (21, 21+128, s)-Nets in Base 9

(21, 21+128, 74)-Net over F₉ — Constructive and digital

Digital (21, 149, 74)-net over F₉, using

t-expansion [i] based on digital (17, 149, 74)-net over F₉, using
- net from sequence [i] based on digital (17, 73)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
    - a function field by GarcÃa and Quoos [i]

(21, 21+128, 88)-Net over F₉ — Digital

Digital (21, 149, 88)-net over F₉, using

net from sequence [i] based on digital (21, 87)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 21 and N(F) ≥ 88, using
  - a curve from the manYPoints database [i]

(21, 21+128, 437)-Net over F₉ — Upper bound on s (digital)

There is no digital (21, 149, 438)-net over F₉, because

2 times m-reduction [i] would yield digital (21, 147, 438)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9¹⁴⁷, 438, F₉, 126) (dual of [438, 291, 127]-code), but
  - residual code [i] would yield OA(9²¹, 311, S₉, 14), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 109 707795 121304 339193 > 9²¹ [i]

(21, 21+128, 465)-Net in Base 9 — Upper bound on s

There is no (21, 149, 466)-net in base 9, because

24 times m-reduction [i] would yield (21, 125, 466)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 192020 205893 151328 802922 969527 222490 548694 151072 094943 272632 768940 918567 543157 999978 924296 413317 686609 815520 964326 335425 > 9¹²⁵ [i]