MinT - Best Known (72, 72+38, s)-Nets in Base 16

Best Known (72, 72+38, s)-Nets in Base 16

(72, 72+38, 596)-Net over F₁₆ — Constructive and digital

Digital (72, 110, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (15, 34, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 9, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
    2. digital (6, 25, 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 (38, 76, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 38, 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]

(72, 72+38, 643)-Net in Base 16 — Constructive

(72, 110, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 34, 129)-net in base 16, using
  - 1 times m-reduction [i] based on (15, 35, 129)-net in base 16, using
    - base change [i] based on digital (0, 20, 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 (38, 76, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 38, 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]

(72, 72+38, 4129)-Net over F₁₆ — Digital

Digital (72, 110, 4129)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁰, 4129, F₁₆, 38) (dual of [4129, 4019, 39]-code), using
- 23 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 18 times 0) [i] based on linear OA(16¹⁰⁷, 4103, F₁₆, 38) (dual of [4103, 3996, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(35) [i] based on
    1. linear OA(16¹⁰⁶, 4096, F₁₆, 38) (dual of [4096, 3990, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(16¹⁰⁰, 4096, F₁₆, 36) (dual of [4096, 3996, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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]

(72, 72+38, 4947027)-Net in Base 16 — Upper bound on s

There is no (72, 110, 4947028)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 839222 488836 406473 944215 985375 594819 909719 996350 669380 040996 428146 920971 038169 646818 079421 099493 971314 373905 372800 513636 032024 921331 > 16¹¹⁰ [i]