MinT - Best Known (73, 73+29, s)-Nets in Base 9

Best Known (73, 73+29, s)-Nets in Base 9

(73, 73+29, 740)-Net over F₉ — Constructive and digital

Digital (73, 102, 740)-net over F₉, using

12 times m-reduction [i] based on digital (73, 114, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 57, 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]

(73, 73+29, 5053)-Net over F₉ — Digital

Digital (73, 102, 5053)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁰², 5053, F₉, 29) (dual of [5053, 4951, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹⁰², 6566, F₉, 29) (dual of [6566, 6464, 30]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(9¹⁰¹, 6565, F₉, 29) (dual of [6565, 6464, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
      1. linear OA(9¹⁰¹, 6561, F₉, 29) (dual of [6561, 6460, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(9⁹⁷, 6561, F₉, 28) (dual of [6561, 6464, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(9⁰, 4, F₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(73, 73+29, 5788222)-Net in Base 9 — Upper bound on s

There is no (73, 102, 5788223)-net in base 9, because

1 times m-reduction [i] would yield (73, 101, 5788223)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 390527 238177 003450 218944 251853 353554 328918 806999 224428 946641 758504 874943 823573 195553 899629 929361 > 9¹⁰¹ [i]