MinT - Best Known (41−13, 41, s)-Nets in Base 16

Best Known (41−13, 41, s)-Nets in Base 16

(41−13, 41, 1030)-Net over F₁₆ — Constructive and digital

Digital (28, 41, 1030)-net over F₁₆, using

16¹ times duplication [i] based on digital (27, 40, 1030)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 12, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 6, 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]
  2. digital (15, 28, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 14, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(41−13, 41, 4777)-Net over F₁₆ — Digital

Digital (28, 41, 4777)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴¹, 4777, F₁₆, 13) (dual of [4777, 4736, 14]-code), using
- 674 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 13 times 0, 1, 137 times 0, 1, 521 times 0) [i] based on linear OA(16³⁷, 4099, F₁₆, 13) (dual of [4099, 4062, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
    1. linear OA(16³⁷, 4096, F₁₆, 13) (dual of [4096, 4059, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(16³⁴, 4096, F₁₆, 12) (dual of [4096, 4062, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [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]

(41−13, 41, large)-Net in Base 16 — Upper bound on s

There is no (28, 41, large)-net in base 16, because

11 times m-reduction [i] would yield (28, 30, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]