MinT - Best Known (30, 30+17, s)-Nets in Base 16

Best Known (30, 30+17, s)-Nets in Base 16

(30, 30+17, 563)-Net over F₁₆ — Constructive and digital

Digital (30, 47, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 13, 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 (17, 34, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 17, 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]

(30, 30+17, 579)-Net in Base 16 — Constructive

(30, 47, 579)-net in base 16, using

16¹ times duplication [i] based on (29, 46, 579)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (4, 12, 65)-net in base 16, using
    - base change [i] based on digital (0, 8, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (17, 34, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 17, 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]

(30, 30+17, 2103)-Net over F₁₆ — Digital

Digital (30, 47, 2103)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁷, 2103, F₁₆, 17) (dual of [2103, 2056, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁴⁷, 4100, F₁₆, 17) (dual of [4100, 4053, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(14) [i] based on
    1. linear OA(16⁴⁶, 4096, F₁₆, 17) (dual of [4096, 4050, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(16⁴³, 4096, F₁₆, 15) (dual of [4096, 4053, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(16¹, 4, F₁₆, 1) (dual of [4, 3, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(30, 30+17, 2105173)-Net in Base 16 — Upper bound on s

There is no (30, 47, 2105174)-net in base 16, because

1 times m-reduction [i] would yield (30, 46, 2105174)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 24 519984 227235 117820 983183 859958 550952 422151 436194 166506 > 16⁴⁶ [i]