MinT - Best Known (172, 172+21, s)-Nets in Base 4

Best Known (172, 172+21, s)-Nets in Base 4

(172, 172+21, 838870)-Net over F₄ — Constructive and digital

Digital (172, 193, 838870)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 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 (160, 181, 838860)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁸¹, 838860, F₄, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(4¹⁸¹, 8388601, F₄, 21) (dual of [8388601, 8388420, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(172, 172+21, 4194311)-Net over F₄ — Digital

Digital (172, 193, 4194311)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁹³, 4194311, F₄, 2, 21) (dual of [(4194311, 2), 8388429, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(4¹², 10, F₄, 2, 10) (dual of [(10, 2), 8, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,9P) [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(4¹⁸¹, 4194301, F₄, 2, 21) (dual of [(4194301, 2), 8388421, 22]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4¹⁸¹, 8388602, F₄, 21) (dual of [8388602, 8388421, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(172, 172+21, large)-Net in Base 4 — Upper bound on s

There is no (172, 193, large)-net in base 4, because

19 times m-reduction [i] would yield (172, 174, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]