MinT - Best Known (101−36, 101, s)-Nets in Base 16

Best Known (101−36, 101, s)-Nets in Base 16

(101−36, 101, 583)-Net over F₁₆ — Constructive and digital

Digital (65, 101, 583)-net over F₁₆, using

1 times m-reduction [i] based on digital (65, 102, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 24, 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 (41, 78, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(101−36, 101, 594)-Net in Base 16 — Constructive

(65, 101, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (11, 29, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (11, 30, 80)-net in base 16, using
    - base change [i] based on digital (1, 20, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (36, 72, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 36, 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]

(101−36, 101, 3122)-Net over F₁₆ — Digital

Digital (65, 101, 3122)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰¹, 3122, F₁₆, 36) (dual of [3122, 3021, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹⁰¹, 4103, F₁₆, 36) (dual of [4103, 4002, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(33) [i] based on
    1. 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]
    2. linear OA(16⁹⁴, 4096, F₁₆, 34) (dual of [4096, 4002, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(101−36, 101, 2874010)-Net in Base 16 — Upper bound on s

There is no (65, 101, 2874011)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 316135 687371 576933 660227 741451 404672 805112 864980 990690 588658 933620 709826 282673 151347 356978 617206 792954 458439 192386 117771 > 16¹⁰¹ [i]