MinT - Best Known (66−21, 66, s)-Nets in Base 16

Best Known (66−21, 66, s)-Nets in Base 16

(66−21, 66, 1032)-Net over F₁₆ — Constructive and digital

Digital (45, 66, 1032)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (12, 22, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 11, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (23, 44, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 22, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(66−21, 66, 5253)-Net over F₁₆ — Digital

Digital (45, 66, 5253)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁶, 5253, F₁₆, 21) (dual of [5253, 5187, 22]-code), using
- 1146 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 15 times 0, 1, 56 times 0, 1, 158 times 0, 1, 343 times 0, 1, 565 times 0) [i] based on linear OA(16⁵⁸, 4099, F₁₆, 21) (dual of [4099, 4041, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
    1. linear OA(16⁵⁸, 4096, F₁₆, 21) (dual of [4096, 4038, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(16⁵⁵, 4096, F₁₆, 20) (dual of [4096, 4041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(66−21, 66, large)-Net in Base 16 — Upper bound on s

There is no (45, 66, large)-net in base 16, because

19 times m-reduction [i] would yield (45, 47, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]