MinT - Best Known (33, 33+15, s)-Nets in Base 16

Best Known (33, 33+15, s)-Nets in Base 16

(33, 33+15, 1032)-Net over F₁₆ — Constructive and digital

Digital (33, 48, 1032)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 8, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (17, 32, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 16, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(33, 33+15, 5458)-Net over F₁₆ — Digital

Digital (33, 48, 5458)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁸, 5458, F₁₆, 15) (dual of [5458, 5410, 16]-code), using
- 1354 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 13 times 0, 1, 113 times 0, 1, 405 times 0, 1, 819 times 0) [i] based on linear OA(16⁴³, 4099, F₁₆, 15) (dual of [4099, 4056, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
    1. linear OA(16⁴³, 4096, F₁₆, 15) (dual of [4096, 4053, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    2. linear OA(16⁴⁰, 4096, F₁₆, 14) (dual of [4096, 4056, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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]

(33, 33+15, large)-Net in Base 16 — Upper bound on s

There is no (33, 48, large)-net in base 16, because

13 times m-reduction [i] would yield (33, 35, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]