MinT - Best Known (90, 111, s)-Nets in Base 16

Best Known (90, 111, s)-Nets in Base 16

(90, 111, 104907)-Net over F₁₆ — Constructive and digital

Digital (90, 111, 104907)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (75, 96, 104858)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁶, 104858, F₁₆, 21, 21) (dual of [(104858, 21), 2201922, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(16⁹⁶, 1048581, F₁₆, 21) (dual of [1048581, 1048485, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(16⁹⁶, 1048576, F₁₆, 21) (dual of [1048576, 1048480, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(16⁹¹, 1048576, F₁₆, 20) (dual of [1048576, 1048485, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 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]

(90, 111, 209716)-Net in Base 16 — Constructive

(90, 111, 209716)-net in base 16, using

16¹ times duplication [i] based on (89, 110, 209716)-net in base 16, using
- net defined by OOA [i] based on OOA(16¹¹⁰, 209716, S₁₆, 21, 21), using
  - OOA 10-folding and stacking with additional row [i] based on OA(16¹¹⁰, 2097161, S₁₆, 21), using
    - 1 times code embedding in larger space [i] based on OA(16¹⁰⁹, 2097160, S₁₆, 21), using
      - discarding parts of the base [i] based on linear OA(128⁶², 2097160, F₁₂₈, 21) (dual of [2097160, 2097098, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(128⁶¹, 2097153, F₁₂₈, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(128⁵⁵, 2097153, F₁₂₈, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(128¹, 7, F₁₂₈, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(90, 111, 2667365)-Net over F₁₆ — Digital

Digital (90, 111, 2667365)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 111, large)-Net in Base 16 — Upper bound on s

There is no (90, 111, large)-net in base 16, because

19 times m-reduction [i] would yield (90, 92, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (90, 111, s)-Nets in Base 16

(90, 111, 104907)-Net over F16 — Constructive and digital

(90, 111, 209716)-Net in Base 16 — Constructive

(90, 111, 2667365)-Net over F16 — Digital

(90, 111, large)-Net in Base 16 — Upper bound on s

(90, 111, 104907)-Net over F₁₆ — Constructive and digital

(90, 111, 2667365)-Net over F₁₆ — Digital