MinT - Best Known (82−16, 82, s)-Nets in Base 9

Best Known (82−16, 82, s)-Nets in Base 9

(82−16, 82, 7409)-Net over F₉ — Constructive and digital

Digital (66, 82, 7409)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (55, 71, 7381)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁷¹, 7381, F₉, 16, 16) (dual of [(7381, 16), 118025, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(9⁷¹, 59048, F₉, 16) (dual of [59048, 58977, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁷¹, 59049, F₉, 16) (dual of [59049, 58978, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(82−16, 82, 132199)-Net over F₉ — Digital

Digital (66, 82, 132199)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−16, 82, large)-Net in Base 9 — Upper bound on s

There is no (66, 82, large)-net in base 9, because

14 times m-reduction [i] would yield (66, 68, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]