MinT - Best Known (252−26, 252, s)-Nets in Base 4

Best Known (252−26, 252, s)-Nets in Base 4

(252−26, 252, 645304)-Net over F₄ — Constructive and digital

Digital (226, 252, 645304)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 23, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, 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]

(252−26, 252, 6468213)-Net over F₄ — Digital

Digital (226, 252, 6468213)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵², 6468213, F₄, 26) (dual of [6468213, 6467961, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵², large, F₄, 26) (dual of [large, large−252, 27]-code), using
  - strength reduction [i] based on linear OA(4²⁵², large, F₄, 28) (dual of [large, large−252, 29]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(252−26, 252, large)-Net in Base 4 — Upper bound on s

There is no (226, 252, large)-net in base 4, because

24 times m-reduction [i] would yield (226, 228, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]