MinT - Best Known (227−29, 227, s)-Nets in Base 4

Best Known (227−29, 227, s)-Nets in Base 4

(227−29, 227, 74908)-Net over F₄ — Constructive and digital

Digital (198, 227, 74908)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 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 (182, 211, 74898)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²¹¹, 74898, F₄, 29, 29) (dual of [(74898, 29), 2171831, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(4²¹¹, 1048573, F₄, 29) (dual of [1048573, 1048362, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²¹¹, 1048576, F₄, 29) (dual of [1048576, 1048365, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(227−29, 227, 524326)-Net over F₄ — Digital

Digital (198, 227, 524326)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²²⁷, 524326, F₄, 2, 29) (dual of [(524326, 2), 1048425, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4²²⁷, 1048652, F₄, 29) (dual of [1048652, 1048425, 30]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²²⁶, 1048651, F₄, 29) (dual of [1048651, 1048425, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(20) [i] based on
      1. linear OA(4²¹¹, 1048576, F₄, 29) (dual of [1048576, 1048365, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(4¹⁵¹, 1048576, F₄, 21) (dual of [1048576, 1048425, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(4¹⁵, 75, F₄, 7) (dual of [75, 60, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁵, 85, F₄, 7) (dual of [85, 70, 8]-code), using
        a code from Grasslâ€™s database [i]

(227−29, 227, large)-Net in Base 4 — Upper bound on s

There is no (198, 227, large)-net in base 4, because

27 times m-reduction [i] would yield (198, 200, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]