MinT - Best Known (156−17, 156, s)-Nets in Base 4

Best Known (156−17, 156, s)-Nets in Base 4

(156−17, 156, 1048589)-Net over F₄ — Constructive and digital

Digital (139, 156, 1048589)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 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 (128, 145, 1048575)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁴⁵, 1048575, F₄, 17, 17) (dual of [(1048575, 17), 17825630, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(4¹⁴⁵, 8388601, F₄, 17) (dual of [8388601, 8388456, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(156−17, 156, 4194315)-Net over F₄ — Digital

Digital (139, 156, 4194315)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁵⁶, 4194315, F₄, 2, 17) (dual of [(4194315, 2), 8388474, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(4¹¹, 14, F₄, 2, 8) (dual of [(14, 2), 17, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,19P) [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(4¹⁴⁵, 4194301, F₄, 2, 17) (dual of [(4194301, 2), 8388457, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4¹⁴⁵, 8388602, F₄, 17) (dual of [8388602, 8388457, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(156−17, 156, large)-Net in Base 4 — Upper bound on s

There is no (139, 156, large)-net in base 4, because

15 times m-reduction [i] would yield (139, 141, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]