MinT - Best Known (109−22, 109, s)-Nets in Base 16

Best Known (109−22, 109, s)-Nets in Base 16

(109−22, 109, 95328)-Net over F₁₆ — Constructive and digital

Digital (87, 109, 95328)-net over F₁₆, using

16¹ times duplication [i] based on digital (86, 108, 95328)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16¹⁰⁸, 95328, F₁₆, 22, 22) (dual of [(95328, 22), 2097108, 23]-NRT-code), using
  - OA 11-folding and stacking [i] based on linear OA(16¹⁰⁸, 1048608, F₁₆, 22) (dual of [1048608, 1048500, 23]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(16¹⁰⁷, 1048607, F₁₆, 22) (dual of [1048607, 1048500, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
        linear OA(16¹⁰¹, 1048576, F₁₆, 22) (dual of [1048576, 1048475, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(16⁷⁶, 1048576, F₁₆, 17) (dual of [1048576, 1048500, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(16⁶, 31, F₁₆, 4) (dual of [31, 25, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁶, 240, F₁₆, 4) (dual of [240, 234, 5]-code), using
        1 times truncation [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]

(109−22, 109, 1048611)-Net over F₁₆ — Digital

Digital (87, 109, 1048611)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰⁹, 1048611, F₁₆, 22) (dual of [1048611, 1048502, 23]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(16¹⁰⁷, 1048607, F₁₆, 22) (dual of [1048607, 1048500, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
      1. linear OA(16¹⁰¹, 1048576, F₁₆, 22) (dual of [1048576, 1048475, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(16⁷⁶, 1048576, F₁₆, 17) (dual of [1048576, 1048500, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      3. linear OA(16⁶, 31, F₁₆, 4) (dual of [31, 25, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁶, 240, F₁₆, 4) (dual of [240, 234, 5]-code), using
        1 times truncation [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]
  2. linear OA(16¹⁰⁷, 1048609, F₁₆, 21) (dual of [1048609, 1048502, 22]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹⁰⁷Â > V_b^sâˆ’1(kâˆ’1)Â = 353090 277226 406078 116575 286121 936373 062540 042872 065185 035191 613776 459335 310062 216331 505724 226055 090467 480985 355830 227884 218171 [i]
  3. linear OA(16⁰, 2, F₁₆, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(109−22, 109, large)-Net in Base 16 — Upper bound on s

There is no (87, 109, large)-net in base 16, because

20 times m-reduction [i] would yield (87, 89, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (109−22, 109, s)-Nets in Base 16

(109−22, 109, 95328)-Net over F16 — Constructive and digital

(109−22, 109, 1048611)-Net over F16 — Digital

(109−22, 109, large)-Net in Base 16 — Upper bound on s

(109−22, 109, 95328)-Net over F₁₆ — Constructive and digital

(109−22, 109, 1048611)-Net over F₁₆ — Digital