MinT - Best Known (144−31, 144, s)-Nets in Base 9

Best Known (144−31, 144, s)-Nets in Base 9

(144−31, 144, 3938)-Net over F₉ — Constructive and digital

Digital (113, 144, 3938)-net over F₉, using

9² times duplication [i] based on digital (111, 142, 3938)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁴², 3938, F₉, 31, 31) (dual of [(3938, 31), 121936, 32]-NRT-code), using
  - OOA 15-folding and stacking with additional row [i] based on linear OA(9¹⁴², 59071, F₉, 31) (dual of [59071, 58929, 32]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹⁴², 59075, F₉, 31) (dual of [59075, 58933, 32]-code), using
      - constructionÂ X applied to C_e(30) âŠ‚ C_e(25) [i] based on
        linear OA(9¹³⁶, 59049, F₉, 31) (dual of [59049, 58913, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
        linear OA(9¹¹⁶, 59049, F₉, 26) (dual of [59049, 58933, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
        linear OA(9⁶, 26, F₉, 4) (dual of [26, 20, 5]-code), using
        discarding factors / shortening the dual code 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]

(144−31, 144, 59083)-Net over F₉ — Digital

Digital (113, 144, 59083)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁴⁴, 59083, F₉, 31) (dual of [59083, 58939, 32]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9¹⁴³, 59081, F₉, 31) (dual of [59081, 58938, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(24) [i] based on
      1. linear OA(9¹³⁶, 59049, F₉, 31) (dual of [59049, 58913, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(9¹¹¹, 59049, F₉, 25) (dual of [59049, 58938, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      3. linear OA(9⁷, 32, F₉, 5) (dual of [32, 25, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁷, 73, F₉, 5) (dual of [73, 66, 6]-code), using
        a code by Danev and Olsson [i]
  2. linear OA(9¹⁴³, 59082, F₉, 30) (dual of [59082, 58939, 31]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9¹⁴³Â > V_b^sâˆ’1(kâˆ’1)Â = 40 935474 945383 310002 397984 639599 031806 394981 854493 529621 195622 042944 633986 384740 158498 823430 734310 111359 087917 914915 861979 741799 178057 [i]
  3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 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]

(144−31, 144, large)-Net in Base 9 — Upper bound on s

There is no (113, 144, large)-net in base 9, because

29 times m-reduction [i] would yield (113, 115, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (144−31, 144, s)-Nets in Base 9

(144−31, 144, 3938)-Net over F9 — Constructive and digital

(144−31, 144, 59083)-Net over F9 — Digital

(144−31, 144, large)-Net in Base 9 — Upper bound on s

(144−31, 144, 3938)-Net over F₉ — Constructive and digital

(144−31, 144, 59083)-Net over F₉ — Digital