MinT - Best Known (78−25, 78, s)-Nets in Base 16

Best Known (78−25, 78, s)-Nets in Base 16

(78−25, 78, 1032)-Net over F₁₆ — Constructive and digital

Digital (53, 78, 1032)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (14, 26, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 13, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (27, 52, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 26, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(78−25, 78, 5369)-Net over F₁₆ — Digital

Digital (53, 78, 5369)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁸, 5369, F₁₆, 25) (dual of [5369, 5291, 26]-code), using
- 1262 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 16 times 0, 1, 64 times 0, 1, 200 times 0, 1, 411 times 0, 1, 562 times 0) [i] based on linear OA(16⁷⁰, 4099, F₁₆, 25) (dual of [4099, 4029, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(23) [i] based on
    1. linear OA(16⁷⁰, 4096, F₁₆, 25) (dual of [4096, 4026, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(16⁶⁷, 4096, F₁₆, 24) (dual of [4096, 4029, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [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]

(78−25, 78, large)-Net in Base 16 — Upper bound on s

There is no (53, 78, large)-net in base 16, because

23 times m-reduction [i] would yield (53, 55, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]