MinT - Best Known (191, 191+24, s)-Nets in Base 4

Best Known (191, 191+24, s)-Nets in Base 4

(191, 191+24, 349542)-Net over F₄ — Constructive and digital

Digital (191, 215, 349542)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 17, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (174, 198, 349525)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹⁸, 349525, F₄, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(4¹⁹⁸, 4194300, F₄, 24) (dual of [4194300, 4194102, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹⁸, 4194304, F₄, 24) (dual of [4194304, 4194106, 25]-code), using
        1 times truncation [i] based on linear OA(4¹⁹⁹, 4194305, F₄, 25) (dual of [4194305, 4194106, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(191, 191+24, 2168353)-Net over F₄ — Digital

Digital (191, 215, 2168353)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁵, 2168353, F₄, 24) (dual of [2168353, 2168138, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹⁵, 4194387, F₄, 24) (dual of [4194387, 4194172, 25]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(4¹⁹⁹, 4194305, F₄, 25) (dual of [4194305, 4194106, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(4¹³³, 4194305, F₄, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [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]

(191, 191+24, large)-Net in Base 4 — Upper bound on s

There is no (191, 215, large)-net in base 4, because

22 times m-reduction [i] would yield (191, 193, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]