MinT - Best Known (47, 72, s)-Nets in Base 16

Best Known (47, 72, s)-Nets in Base 16

(47, 72, 583)-Net over F₁₆ — Constructive and digital

Digital (47, 72, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 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 (29, 54, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 27, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(47, 72, 643)-Net in Base 16 — Constructive

(47, 72, 643)-net in base 16, using

16¹ times duplication [i] based on (46, 71, 643)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (9, 21, 129)-net in base 16, using
    - base change [i] based on digital (0, 12, 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 (25, 50, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 25, 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]

(47, 72, 3265)-Net over F₁₆ — Digital

Digital (47, 72, 3265)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷², 3265, F₁₆, 25) (dual of [3265, 3193, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷², 4107, F₁₆, 25) (dual of [4107, 4035, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
    1. linear OA(16⁷⁰, 4096, F₁₆, 25) (dual of [4096, 4026, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(16⁶¹, 4096, F₁₆, 22) (dual of [4096, 4035, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(16², 11, F₁₆, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
      - discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]

(47, 72, 4695113)-Net in Base 16 — Upper bound on s

There is no (47, 72, 4695114)-net in base 16, because

1 times m-reduction [i] would yield (47, 71, 4695114)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 082707 190643 568897 300909 733240 551283 630071 375529 187591 422977 672945 580235 814612 802896 > 16⁷¹ [i]