MinT - Best Known (141−25, 141, s)-Nets in Base 4

Best Known (141−25, 141, s)-Nets in Base 4

(141−25, 141, 1375)-Net over F₄ — Constructive and digital

Digital (116, 141, 1375)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 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 (102, 127, 1365)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹²⁷, 1365, F₄, 25, 25) (dual of [(1365, 25), 33998, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(4¹²⁷, 16381, F₄, 25) (dual of [16381, 16254, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹²⁷, 16384, F₄, 25) (dual of [16384, 16257, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(141−25, 141, 14505)-Net over F₄ — Digital

Digital (116, 141, 14505)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴¹, 14505, F₄, 25) (dual of [14505, 14364, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁴¹, 16433, F₄, 25) (dual of [16433, 16292, 26]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁴⁰, 16432, F₄, 25) (dual of [16432, 16292, 26]-code), using
    - constructionÂ X applied to C_e(24) âŠ‚ C_e(17) [i] based on
      1. linear OA(4¹²⁷, 16384, F₄, 25) (dual of [16384, 16257, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      2. linear OA(4⁹², 16384, F₄, 18) (dual of [16384, 16292, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      3. linear OA(4¹³, 48, F₄, 6) (dual of [48, 35, 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]

(141−25, 141, large)-Net in Base 4 — Upper bound on s

There is no (116, 141, large)-net in base 4, because

23 times m-reduction [i] would yield (116, 118, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]