MinT - Best Known (64, 89, s)-Nets in Base 9

Best Known (64, 89, s)-Nets in Base 9

(64, 89, 740)-Net over F₉ — Constructive and digital

Digital (64, 89, 740)-net over F₉, using

7 times m-reduction [i] based on digital (64, 96, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 48, 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]

(64, 89, 5264)-Net over F₉ — Digital

Digital (64, 89, 5264)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, 5264, F₉, 25) (dual of [5264, 5175, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁸⁹, 6561, F₉, 25) (dual of [6561, 6472, 26]-code), using
  - an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(64, 89, 6577325)-Net in Base 9 — Upper bound on s

There is no (64, 89, 6577326)-net in base 9, because

1 times m-reduction [i] would yield (64, 88, 6577326)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 940461 108397 738608 900648 936910 484040 341960 018788 945971 277579 223095 609926 150399 348161 > 9⁸⁸ [i]