MinT - Best Known (60−19, 60, s)-Nets in Base 16

Best Known (60−19, 60, s)-Nets in Base 16

(60−19, 60, 1032)-Net over F₁₆ — Constructive and digital

Digital (41, 60, 1032)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 10, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (21, 40, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 20, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(60−19, 60, 5262)-Net over F₁₆ — Digital

Digital (41, 60, 5262)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁰, 5262, F₁₆, 19) (dual of [5262, 5202, 20]-code), using
- 1155 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 151 times 0, 1, 337 times 0, 1, 589 times 0) [i] based on linear OA(16⁵², 4099, F₁₆, 19) (dual of [4099, 4047, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
    1. linear OA(16⁵², 4096, F₁₆, 19) (dual of [4096, 4044, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(16⁴⁹, 4096, F₁₆, 18) (dual of [4096, 4047, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(60−19, 60, large)-Net in Base 16 — Upper bound on s

There is no (41, 60, large)-net in base 16, because

17 times m-reduction [i] would yield (41, 43, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]