MinT - Best Known (206−22, 206, s)-Nets in Base 4

Best Known (206−22, 206, s)-Nets in Base 4

(206−22, 206, 762610)-Net over F₄ — Constructive and digital

Digital (184, 206, 762610)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 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 (171, 193, 762600)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹³, 762600, F₄, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(4¹⁹³, 8388600, F₄, 22) (dual of [8388600, 8388407, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹³, large, F₄, 22) (dual of [large, large−193, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(206−22, 206, 4194311)-Net over F₄ — Digital

Digital (184, 206, 4194311)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁰⁶, 4194311, F₄, 2, 22) (dual of [(4194311, 2), 8388416, 23]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(4¹³, 10, F₄, 2, 11) (dual of [(10, 2), 7, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,8P) [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(4¹⁹³, 4194301, F₄, 2, 22) (dual of [(4194301, 2), 8388409, 23]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4¹⁹³, 8388602, F₄, 22) (dual of [8388602, 8388409, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹³, large, F₄, 22) (dual of [large, large−193, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(206−22, 206, large)-Net in Base 4 — Upper bound on s

There is no (184, 206, large)-net in base 4, because

20 times m-reduction [i] would yield (184, 186, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]