MinT - Best Known (216, 216+34, s)-Nets in Base 4

Best Known (216, 216+34, s)-Nets in Base 4

(216, 216+34, 15441)-Net over F₄ — Constructive and digital

Digital (216, 250, 15441)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (192, 226, 15420)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²²⁶, 15420, F₄, 34, 34) (dual of [(15420, 34), 524054, 35]-NRT-code), using
    - OA 17-folding and stacking [i] based on linear OA(4²²⁶, 262140, F₄, 34) (dual of [262140, 261914, 35]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²²⁶, 262144, F₄, 34) (dual of [262144, 261918, 35]-code), using
        an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(216, 216+34, 206293)-Net over F₄ — Digital

Digital (216, 250, 206293)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁰, 206293, F₄, 34) (dual of [206293, 206043, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁰, 262231, F₄, 34) (dual of [262231, 261981, 35]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(4²⁴⁷, 262228, F₄, 34) (dual of [262228, 261981, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(24) [i] based on
      1. linear OA(4²²⁶, 262144, F₄, 34) (dual of [262144, 261918, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(4¹⁶³, 262144, F₄, 25) (dual of [262144, 261981, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      3. linear OA(4²¹, 84, F₄, 8) (dual of [84, 63, 9]-code), using
        1 times truncation [i] based on linear OA(4²², 85, F₄, 9) (dual of [85, 63, 10]-code), using
        the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {−20,−17,−14,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(216, 216+34, large)-Net in Base 4 — Upper bound on s

There is no (216, 250, large)-net in base 4, because

32 times m-reduction [i] would yield (216, 218, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]