MinT - Best Known (191, 216, s)-Nets in Base 4

Best Known (191, 216, s)-Nets in Base 4

(191, 216, 349542)-Net over F₄ — Constructive and digital

Digital (191, 216, 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, 199, 349525)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹⁹, 349525, F₄, 25, 25) (dual of [(349525, 25), 8737926, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(4¹⁹⁹, 4194301, F₄, 25) (dual of [4194301, 4194102, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹⁹, 4194304, F₄, 25) (dual of [4194304, 4194105, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(191, 216, 2097194)-Net over F₄ — Digital

Digital (191, 216, 2097194)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²¹⁶, 2097194, F₄, 2, 25) (dual of [(2097194, 2), 4194172, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4²¹⁴, 2097193, F₄, 2, 25) (dual of [(2097193, 2), 4194172, 26]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4²¹⁴, 4194386, F₄, 25) (dual of [4194386, 4194172, 26]-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(4¹⁵, 81, F₄, 7) (dual of [81, 66, 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]

(191, 216, large)-Net in Base 4 — Upper bound on s

There is no (191, 216, large)-net in base 4, because

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