MinT - Best Known (85, 118, s)-Nets in Base 9

Best Known (85, 118, s)-Nets in Base 9

(85, 118, 770)-Net over F₉ — Constructive and digital

Digital (85, 118, 770)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 20, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [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]

(85, 118, 6182)-Net over F₉ — Digital

Digital (85, 118, 6182)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹⁸, 6182, F₉, 33) (dual of [6182, 6064, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹¹⁸, 6570, F₉, 33) (dual of [6570, 6452, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
    1. 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]
    2. linear OA(9¹⁰⁹, 6561, F₉, 31) (dual of [6561, 6452, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(9¹, 9, F₉, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(85, 118, 8078794)-Net in Base 9 — Upper bound on s

There is no (85, 118, 8078795)-net in base 9, because

1 times m-reduction [i] would yield (85, 117, 8078795)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4429 694519 806134 002521 736661 647842 213724 166395 018050 911928 912044 194258 672912 194951 152358 913539 889136 641776 116097 > 9¹¹⁷ [i]