MinT - Best Known (84, 117, s)-Nets in Base 9

Best Known (84, 117, s)-Nets in Base 9

(84, 117, 768)-Net over F₉ — Constructive and digital

Digital (84, 117, 768)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 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 (65, 98, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 49, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(84, 117, 5758)-Net over F₉ — Digital

Digital (84, 117, 5758)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹⁷, 5758, F₉, 33) (dual of [5758, 5641, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹¹⁷, 6561, F₉, 33) (dual of [6561, 6444, 34]-code), using
  - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(84, 117, 7042167)-Net in Base 9 — Upper bound on s

There is no (84, 117, 7042168)-net in base 9, because

1 times m-reduction [i] would yield (84, 116, 7042168)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 492 188742 561882 452931 941580 259366 497536 561716 815145 390299 792525 037195 449196 541880 780151 648214 454250 380833 752065 > 9¹¹⁶ [i]