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

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

(30, 30+16, 579)-Net over F₁₆ — Constructive and digital

Digital (30, 46, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 14, 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 (16, 32, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 16, 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+16, 643)-Net in Base 16 — Constructive

(30, 46, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (6, 14, 129)-net in base 16, using
  - base change [i] based on digital (0, 8, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]
2. digital (16, 32, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 16, 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+16, 2984)-Net over F₁₆ — Digital

Digital (30, 46, 2984)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁶, 2984, F₁₆, 16) (dual of [2984, 2938, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁴⁶, 4099, F₁₆, 16) (dual of [4099, 4053, 17]-code), using
  - 1 times truncation [i] 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+16, 2105173)-Net in Base 16 — Upper bound on s

There is no (30, 46, 2105174)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 24 519984 227235 117820 983183 859958 550952 422151 436194 166506 > 16⁴⁶ [i]