MinT - Best Known (105−20, 105, s)-Nets in Base 16

Best Known (105−20, 105, s)-Nets in Base 16

(105−20, 105, 104903)-Net over F₁₆ — Constructive and digital

Digital (85, 105, 104903)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (71, 91, 104858)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹¹, 104858, F₁₆, 20, 20) (dual of [(104858, 20), 2097069, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(16⁹¹, 1048580, F₁₆, 20) (dual of [1048580, 1048489, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁹¹, 1048581, F₁₆, 20) (dual of [1048581, 1048490, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
        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(16⁸⁶, 1048576, F₁₆, 19) (dual of [1048576, 1048490, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(105−20, 105, 209716)-Net in Base 16 — Constructive

(85, 105, 209716)-net in base 16, using

base change [i] based on digital (40, 60, 209716)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128⁶⁰, 209716, F₁₂₈, 20, 20) (dual of [(209716, 20), 4194260, 21]-NRT-code), using
  - OA 10-folding and stacking [i] based on linear OA(128⁶⁰, 2097160, F₁₂₈, 20) (dual of [2097160, 2097100, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(128⁶⁰, 2097163, F₁₂₈, 20) (dual of [2097163, 2097103, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(16) [i] based on
        linear OA(128⁵⁸, 2097152, F₁₂₈, 20) (dual of [2097152, 2097094, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(128⁴⁹, 2097152, F₁₂₈, 17) (dual of [2097152, 2097103, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(128², 11, F₁₂₈, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
        discarding factors / shortening the dual code based on linear OA(128², 128, F₁₂₈, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
        Reedâ€“Solomon code RS(126,128) [i]

(105−20, 105, 2384917)-Net over F₁₆ — Digital

Digital (85, 105, 2384917)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−20, 105, large)-Net in Base 16 — Upper bound on s

There is no (85, 105, large)-net in base 16, because

18 times m-reduction [i] would yield (85, 87, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (105−20, 105, s)-Nets in Base 16

(105−20, 105, 104903)-Net over F16 — Constructive and digital

(105−20, 105, 209716)-Net in Base 16 — Constructive

(105−20, 105, 2384917)-Net over F16 — Digital

(105−20, 105, large)-Net in Base 16 — Upper bound on s

(105−20, 105, 104903)-Net over F₁₆ — Constructive and digital

(105−20, 105, 2384917)-Net over F₁₆ — Digital