MinT - Best Known (61, 95, s)-Nets in Base 16

Best Known (61, 95, s)-Nets in Base 16

(61, 95, 583)-Net over F₁₆ — Constructive and digital

Digital (61, 95, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 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, 72, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 36, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(61, 95, 594)-Net in Base 16 — Constructive

(61, 95, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (10, 27, 80)-net in base 16, using
  - base change [i] based on digital (1, 18, 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 (34, 68, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 34, 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]

(61, 95, 2920)-Net over F₁₆ — Digital

Digital (61, 95, 2920)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁵, 2920, F₁₆, 34) (dual of [2920, 2825, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁵, 4100, F₁₆, 34) (dual of [4100, 4005, 35]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(16⁹⁴, 4099, F₁₆, 34) (dual of [4099, 4005, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
      1. 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]
      2. linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(61, 95, 2563056)-Net in Base 16 — Upper bound on s

There is no (61, 95, 2563057)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 462633 005694 749836 365400 789528 586815 191069 430021 892754 456768 677248 693711 008717 667493 692177 490776 062690 314834 216336 > 16⁹⁵ [i]