MinT - Best Known (111−20, 111, s)-Nets in Base 9

Best Known (111−20, 111, s)-Nets in Base 9

(111−20, 111, 53147)-Net over F₉ — Constructive and digital

Digital (91, 111, 53147)-net over F₉, using

9² times duplication [i] based on digital (89, 109, 53147)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁰⁹, 53147, F₉, 20, 20) (dual of [(53147, 20), 1062831, 21]-NRT-code), using
  - OA 10-folding and stacking [i] based on linear OA(9¹⁰⁹, 531470, F₉, 20) (dual of [531470, 531361, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹⁰⁹, 531471, F₉, 20) (dual of [531471, 531362, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(14) [i] based on
        linear OA(9¹⁰³, 531441, F₉, 20) (dual of [531441, 531338, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(9⁷⁹, 531441, F₉, 15) (dual of [531441, 531362, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(9⁶, 30, F₉, 4) (dual of [30, 24, 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]

(111−20, 111, 531480)-Net over F₉ — Digital

Digital (91, 111, 531480)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹¹, 531480, F₉, 20) (dual of [531480, 531369, 21]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9¹¹⁰, 531478, F₉, 20) (dual of [531478, 531368, 21]-code), using
    - constructionÂ X applied to C_e(19) âŠ‚ C_e(13) [i] based on
      1. linear OA(9¹⁰³, 531441, F₉, 20) (dual of [531441, 531338, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      2. linear OA(9⁷³, 531441, F₉, 14) (dual of [531441, 531368, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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(9¹¹⁰, 531479, F₉, 19) (dual of [531479, 531369, 20]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9¹¹⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 32 202591 027593 227120 146330 000554 571698 073632 719506 854279 528006 866242 126600 521796 350534 008918 149383 574129 [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]

(111−20, 111, large)-Net in Base 9 — Upper bound on s

There is no (91, 111, large)-net in base 9, because

18 times m-reduction [i] would yield (91, 93, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111−20, 111, s)-Nets in Base 9

(111−20, 111, 53147)-Net over F9 — Constructive and digital

(111−20, 111, 531480)-Net over F9 — Digital

(111−20, 111, large)-Net in Base 9 — Upper bound on s

(111−20, 111, 53147)-Net over F₉ — Constructive and digital

(111−20, 111, 531480)-Net over F₉ — Digital