MinT - Best Known (132−17, 132, s)-Nets in Base 4

Best Known (132−17, 132, s)-Nets in Base 4

(132−17, 132, 131086)-Net over F₄ — Constructive and digital

Digital (115, 132, 131086)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (104, 121, 131072)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹²¹, 131072, F₄, 17, 17) (dual of [(131072, 17), 2228103, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(4¹²¹, 1048577, F₄, 17) (dual of [1048577, 1048456, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(132−17, 132, 524314)-Net over F₄ — Digital

Digital (115, 132, 524314)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹³², 524314, F₄, 2, 17) (dual of [(524314, 2), 1048496, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹³², 1048628, F₄, 17) (dual of [1048628, 1048496, 18]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹³¹, 1048627, F₄, 17) (dual of [1048627, 1048496, 18]-code), using
    - constructionÂ X applied to C([0,8]) âŠ‚ C([0,5]) [i] based on
      1. linear OA(4¹²¹, 1048577, F₄, 17) (dual of [1048577, 1048456, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      2. linear OA(4⁸¹, 1048577, F₄, 11) (dual of [1048577, 1048496, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
      3. linear OA(4¹⁰, 50, F₄, 5) (dual of [50, 40, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(132−17, 132, large)-Net in Base 4 — Upper bound on s

There is no (115, 132, large)-net in base 4, because

15 times m-reduction [i] would yield (115, 117, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]