MinT - Best Known (54, 54+29, s)-Nets in Base 16

Best Known (54, 54+29, s)-Nets in Base 16

(54, 54+29, 583)-Net over F₁₆ — Constructive and digital

Digital (54, 83, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (53, 82, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 20, 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 (33, 62, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 31, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(54, 54+29, 643)-Net in Base 16 — Constructive

(54, 83, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (11, 25, 129)-net in base 16, using
  - base change [i] based on (6, 20, 129)-net in base 32, using
    - 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
      - base change [i] based on digital (0, 15, 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 (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(54, 54+29, 3292)-Net over F₁₆ — Digital

Digital (54, 83, 3292)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸³, 3292, F₁₆, 29) (dual of [3292, 3209, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸³, 4103, F₁₆, 29) (dual of [4103, 4020, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(26) [i] based on
    1. linear OA(16⁸², 4096, F₁₆, 29) (dual of [4096, 4014, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(16⁷⁶, 4096, F₁₆, 27) (dual of [4096, 4020, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 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]

(54, 54+29, 4550598)-Net in Base 16 — Upper bound on s

There is no (54, 83, 4550599)-net in base 16, because

1 times m-reduction [i] would yield (54, 82, 4550599)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 813395 541618 488184 038107 625955 948827 345479 299305 019254 665715 758778 656002 301256 836583 910115 211416 > 16⁸² [i]