MinT - Best Known (177, 177+20, s)-Nets in Base 4

Best Known (177, 177+20, s)-Nets in Base 4

(177, 177+20, 838881)-Net over F₄ — Constructive and digital

Digital (177, 197, 838881)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (160, 180, 838860)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁸⁰, 838860, F₄, 20, 20) (dual of [(838860, 20), 16777020, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(4¹⁸⁰, 8388600, F₄, 20) (dual of [8388600, 8388420, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁸⁰, large, F₄, 20) (dual of [large, large−180, 21]-code), using
        the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(177, 177+20, large)-Net over F₄ — Digital

Digital (177, 197, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁷, large, F₄, 20) (dual of [large, large−197, 21]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4¹⁸⁰, large, F₄, 20) (dual of [large, large−180, 21]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(177, 177+20, large)-Net in Base 4 — Upper bound on s

There is no (177, 197, large)-net in base 4, because

18 times m-reduction [i] would yield (177, 179, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]