MinT - Best Known (257−26, 257, s)-Nets in Base 4

Best Known (257−26, 257, s)-Nets in Base 4

(257−26, 257, 645325)-Net over F₄ — Constructive and digital

Digital (231, 257, 645325)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 28, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 14, 24)-net over F₁₆, using
    - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [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]

(257−26, 257, large)-Net over F₄ — Digital

Digital (231, 257, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁷, large, F₄, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] 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]

(257−26, 257, large)-Net in Base 4 — Upper bound on s

There is no (231, 257, large)-net in base 4, because

24 times m-reduction [i] would yield (231, 233, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]