MinT - Best Known (222−18, 222, s)-Nets in Base 4

Best Known (222−18, 222, s)-Nets in Base 4

(222−18, 222, 3728285)-Net over F₄ — Constructive and digital

Digital (204, 222, 3728285)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 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 (190, 208, 3728268)-net over F₄, using
  - trace code for nets [i] based on digital (34, 52, 932067)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁵², 932067, F₂₅₆, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(256⁵², large, F₂₅₆, 18) (dual of [large, large−52, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(222−18, 222, large)-Net over F₄ — Digital

Digital (204, 222, large)-net over F₄, using

5 times m-reduction [i] based on digital (204, 227, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁷, large, F₄, 23) (dual of [large, large−227, 24]-code), using
  - 22 times code embedding in larger space [i] based on linear OA(4²⁰⁵, large, F₄, 23) (dual of [large, large−205, 24]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(222−18, 222, large)-Net in Base 4 — Upper bound on s

There is no (204, 222, large)-net in base 4, because

16 times m-reduction [i] would yield (204, 206, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]