MinT - Best Known (210−32, 210, s)-Nets in Base 4

Best Known (210−32, 210, s)-Nets in Base 4

(210−32, 210, 4106)-Net over F₄ — Constructive and digital

Digital (178, 210, 4106)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (160, 192, 4096)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹², 4096, F₄, 32, 32) (dual of [(4096, 32), 130880, 33]-NRT-code), using
    - OA 16-folding and stacking [i] based on linear OA(4¹⁹², 65536, F₄, 32) (dual of [65536, 65344, 33]-code), using
      - 1 times truncation [i] based on linear OA(4¹⁹³, 65537, F₄, 33) (dual of [65537, 65344, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(210−32, 210, 62785)-Net over F₄ — Digital

Digital (178, 210, 62785)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁰, 62785, F₄, 32) (dual of [62785, 62575, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹⁰, 65602, F₄, 32) (dual of [65602, 65392, 33]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(4²⁰⁶, 65598, F₄, 32) (dual of [65598, 65392, 33]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(4¹⁹³, 65537, F₄, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(4¹⁴⁵, 65537, F₄, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(4¹³, 61, F₄, 6) (dual of [61, 48, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹³, 63, F₄, 6) (dual of [63, 50, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(210−32, 210, large)-Net in Base 4 — Upper bound on s

There is no (178, 210, large)-net in base 4, because

30 times m-reduction [i] would yield (178, 180, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]