MinT - Best Known (93, 93+37, s)-Nets in Base 9

Best Known (93, 93+37, s)-Nets in Base 9

(93, 93+37, 774)-Net over F₉ — Constructive and digital

Digital (93, 130, 774)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 24, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (69, 106, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 53, 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]

(93, 93+37, 5697)-Net over F₉ — Digital

Digital (93, 130, 5697)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³⁰, 5697, F₉, 37) (dual of [5697, 5567, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹³⁰, 6566, F₉, 37) (dual of [6566, 6436, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
    1. linear OA(9¹²⁹, 6561, F₉, 37) (dual of [6561, 6432, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(9¹²⁵, 6561, F₉, 35) (dual of [6561, 6436, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(9¹, 5, F₉, 1) (dual of [5, 4, 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]

(93, 93+37, 6512943)-Net in Base 9 — Upper bound on s

There is no (93, 130, 6512944)-net in base 9, because

1 times m-reduction [i] would yield (93, 129, 6512944)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1251 076425 847928 341712 156910 653980 408695 636622 291399 485363 064673 974135 790403 016698 870875 894315 842797 312915 278655 602492 342529 > 9¹²⁹ [i]