MinT - Best Known (248−28, 248, s)-Nets in Base 4

Best Known (248−28, 248, s)-Nets in Base 4

(248−28, 248, 299607)-Net over F₄ — Constructive and digital

Digital (220, 248, 299607)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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 (203, 231, 299593)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²³¹, 299593, F₄, 28, 28) (dual of [(299593, 28), 8388373, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(4²³¹, 4194302, F₄, 28) (dual of [4194302, 4194071, 29]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²³¹, 4194303, F₄, 28) (dual of [4194303, 4194072, 29]-code), using
        1 times truncation [i] based on linear OA(4²³², 4194304, F₄, 29) (dual of [4194304, 4194072, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(248−28, 248, 2097193)-Net over F₄ — Digital

Digital (220, 248, 2097193)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁴⁸, 2097193, F₄, 2, 28) (dual of [(2097193, 2), 4194138, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4²⁴⁸, 4194386, F₄, 28) (dual of [4194386, 4194138, 29]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(20) [i] based on
    1. linear OA(4²³², 4194304, F₄, 29) (dual of [4194304, 4194072, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(4¹⁶⁶, 4194304, F₄, 21) (dual of [4194304, 4194138, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(4¹⁶, 82, F₄, 6) (dual of [82, 66, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶, 85, F₄, 6) (dual of [85, 69, 7]-code), using
        the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(248−28, 248, large)-Net in Base 4 — Upper bound on s

There is no (220, 248, large)-net in base 4, because

26 times m-reduction [i] would yield (220, 222, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]