MinT - Best Known (150−26, 150, s)-Nets in Base 9

Best Known (150−26, 150, s)-Nets in Base 9

(150−26, 150, 40884)-Net over F₉ — Constructive and digital

Digital (124, 150, 40884)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁵⁰, 40884, F₉, 26, 26) (dual of [(40884, 26), 1062834, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(9¹⁵⁰, 531492, F₉, 26) (dual of [531492, 531342, 27]-code), using
  - discarding factors / shortening the dual code based on linear OA(9¹⁵⁰, 531494, F₉, 26) (dual of [531494, 531344, 27]-code), using
    - constructionÂ X applied to C_e(25) âŠ‚ C_e(18) [i] based on
      1. linear OA(9¹³⁹, 531441, F₉, 26) (dual of [531441, 531302, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      2. linear OA(9⁹⁷, 531441, F₉, 19) (dual of [531441, 531344, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      3. linear OA(9¹¹, 53, F₉, 6) (dual of [53, 42, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(9¹¹, 80, F₉, 6) (dual of [80, 69, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 9²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(150−26, 150, 676081)-Net over F₉ — Digital

Digital (124, 150, 676081)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(150−26, 150, large)-Net in Base 9 — Upper bound on s

There is no (124, 150, large)-net in base 9, because

24 times m-reduction [i] would yield (124, 126, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]