MinT - Best Known (119−17, 119, s)-Nets in Base 16

Best Known (119−17, 119, s)-Nets in Base 16

(119−17, 119, 2097688)-Net over F₁₆ — Constructive and digital

Digital (102, 119, 2097688)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 538)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
    2. digital (8, 16, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (81, 98, 2097150)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(119−17, 119, 2097697)-Net in Base 16 — Constructive

(102, 119, 2097697)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (13, 21, 547)-net in base 16, using
  - (u,Â u+v)-construction [i] based on
    1. (1, 5, 33)-net in base 16, using
      - base change [i] based on digital (0, 4, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    2. digital (8, 16, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (81, 98, 2097150)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(119−17, 119, large)-Net over F₁₆ — Digital

Digital (102, 119, large)-net over F₁₆, using

t-expansion [i] based on digital (99, 119, large)-net over F₁₆, using
- 2 times m-reduction [i] based on digital (99, 121, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²¹, large, F₁₆, 22) (dual of [large, large−121, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(119−17, 119, large)-Net in Base 16 — Upper bound on s

There is no (102, 119, large)-net in base 16, because

15 times m-reduction [i] would yield (102, 104, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (119−17, 119, s)-Nets in Base 16

(119−17, 119, 2097688)-Net over F16 — Constructive and digital

(119−17, 119, 2097697)-Net in Base 16 — Constructive

(119−17, 119, large)-Net over F16 — Digital

(119−17, 119, large)-Net in Base 16 — Upper bound on s

(119−17, 119, 2097688)-Net over F₁₆ — Constructive and digital

(119−17, 119, large)-Net over F₁₆ — Digital