MinT - Best Known (42−8, 42, s)-Nets in Base 9

Best Known (42−8, 42, s)-Nets in Base 9

(42−8, 42, 14799)-Net over F₉ — Constructive and digital

Digital (34, 42, 14799)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 36)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁶, 36, F₉, 4, 4) (dual of [(36, 4), 138, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(9⁶, 36, F₉, 3, 4) (dual of [(36, 3), 102, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(9⁶, 72, F₉, 4) (dual of [72, 66, 5]-code), using
        1 times truncation [i] based on linear OA(9⁷, 73, F₉, 5) (dual of [73, 66, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (28, 36, 14763)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9³⁶, 14763, F₉, 8, 8) (dual of [(14763, 8), 118068, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(9³⁶, 59052, F₉, 8) (dual of [59052, 59016, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(9³⁶, 59054, F₉, 8) (dual of [59054, 59018, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(9³⁶, 59049, F₉, 8) (dual of [59049, 59013, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(9³¹, 59049, F₉, 7) (dual of [59049, 59018, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(9⁰, 5, F₉, 0) (dual of [5, 5, 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]

(42−8, 42, 224538)-Net over F₉ — Digital

Digital (34, 42, 224538)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−8, 42, large)-Net in Base 9 — Upper bound on s

There is no (34, 42, large)-net in base 9, because

6 times m-reduction [i] would yield (34, 36, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (42−8, 42, s)-Nets in Base 9

(42−8, 42, 14799)-Net over F9 — Constructive and digital

(42−8, 42, 224538)-Net over F9 — Digital

(42−8, 42, large)-Net in Base 9 — Upper bound on s

(42−8, 42, 14799)-Net over F₉ — Constructive and digital

(42−8, 42, 224538)-Net over F₉ — Digital