MinT - Best Known (123−22, 123, s)-Nets in Base 9

Best Known (123−22, 123, s)-Nets in Base 9

(123−22, 123, 48316)-Net over F₉ — Constructive and digital

Digital (101, 123, 48316)-net over F₉, using

9¹ times duplication [i] based on digital (100, 122, 48316)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹²², 48316, F₉, 22, 22) (dual of [(48316, 22), 1062830, 23]-NRT-code), using
  - OA 11-folding and stacking [i] based on linear OA(9¹²², 531476, F₉, 22) (dual of [531476, 531354, 23]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹²², 531478, F₉, 22) (dual of [531478, 531356, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
        linear OA(9¹¹⁵, 531441, F₉, 22) (dual of [531441, 531326, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(9⁸⁵, 531441, F₉, 16) (dual of [531441, 531356, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(9⁷, 37, F₉, 5) (dual of [37, 30, 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]

(123−22, 123, 531480)-Net over F₉ — Digital

Digital (101, 123, 531480)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹²³, 531480, F₉, 22) (dual of [531480, 531357, 23]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9¹²², 531478, F₉, 22) (dual of [531478, 531356, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
      1. linear OA(9¹¹⁵, 531441, F₉, 22) (dual of [531441, 531326, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(9⁸⁵, 531441, F₉, 16) (dual of [531441, 531356, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(9⁷, 37, F₉, 5) (dual of [37, 30, 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(9¹²², 531479, F₉, 21) (dual of [531479, 531357, 22]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9¹²²Â > V_b^sâˆ’1(kâˆ’1)Â = 1 531890 586432 090422 235801 087605 607266 003893 277909 675898 900135 411299 342129 050998 961440 117071 303794 961578 449881 780849 [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]

(123−22, 123, large)-Net in Base 9 — Upper bound on s

There is no (101, 123, large)-net in base 9, because

20 times m-reduction [i] would yield (101, 103, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (123−22, 123, s)-Nets in Base 9

(123−22, 123, 48316)-Net over F9 — Constructive and digital

(123−22, 123, 531480)-Net over F9 — Digital

(123−22, 123, large)-Net in Base 9 — Upper bound on s

(123−22, 123, 48316)-Net over F₉ — Constructive and digital

(123−22, 123, 531480)-Net over F₉ — Digital