MinT - Best Known (244−26, 244, s)-Nets in Base 4

Best Known (244−26, 244, s)-Nets in Base 4

(244−26, 244, 645287)-Net over F₄ — Constructive and digital

Digital (218, 244, 645287)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 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 (203, 229, 645277)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²²⁹, 645277, F₄, 26, 26) (dual of [(645277, 26), 16776973, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(4²²⁹, 8388601, F₄, 26) (dual of [8388601, 8388372, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²²⁹, large, F₄, 26) (dual of [large, large−229, 27]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(244−26, 244, 4194311)-Net over F₄ — Digital

Digital (218, 244, 4194311)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁴⁴, 4194311, F₄, 2, 26) (dual of [(4194311, 2), 8388378, 27]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(4¹⁵, 10, F₄, 2, 13) (dual of [(10, 2), 5, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,6P) [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(4²²⁹, 4194301, F₄, 2, 26) (dual of [(4194301, 2), 8388373, 27]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4²²⁹, 8388602, F₄, 26) (dual of [8388602, 8388373, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²²⁹, large, F₄, 26) (dual of [large, large−229, 27]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(244−26, 244, large)-Net in Base 4 — Upper bound on s

There is no (218, 244, large)-net in base 4, because

24 times m-reduction [i] would yield (218, 220, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]