MinT - Best Known (112−21, 112, s)-Nets in Base 16

Best Known (112−21, 112, s)-Nets in Base 16

(112−21, 112, 104923)-Net over F₁₆ — Constructive and digital

Digital (91, 112, 104923)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [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]

(112−21, 112, 209716)-Net in Base 16 — Constructive

(91, 112, 209716)-net in base 16, using

base change [i] based on digital (43, 64, 209716)-net over F₁₂₈, using
- 128¹ times duplication [i] based on digital (42, 63, 209716)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁶³, 209716, F₁₂₈, 21, 21) (dual of [(209716, 21), 4403973, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁶³, 2097161, F₁₂₈, 21) (dual of [2097161, 2097098, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁶³, 2097163, F₁₂₈, 21) (dual of [2097163, 2097100, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
        linear OA(128⁶¹, 2097152, F₁₂₈, 21) (dual of [2097152, 2097091, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(128⁵², 2097152, F₁₂₈, 18) (dual of [2097152, 2097100, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(112−21, 112, 3063997)-Net over F₁₆ — Digital

Digital (91, 112, 3063997)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−21, 112, large)-Net in Base 16 — Upper bound on s

There is no (91, 112, large)-net in base 16, because

19 times m-reduction [i] would yield (91, 93, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (112−21, 112, s)-Nets in Base 16

(112−21, 112, 104923)-Net over F16 — Constructive and digital

(112−21, 112, 209716)-Net in Base 16 — Constructive

(112−21, 112, 3063997)-Net over F16 — Digital

(112−21, 112, large)-Net in Base 16 — Upper bound on s

(112−21, 112, 104923)-Net over F₁₆ — Constructive and digital

(112−21, 112, 3063997)-Net over F₁₆ — Digital