MinT - Best Known (251−46, 251, s)-Nets in Base 4

Best Known (251−46, 251, s)-Nets in Base 4

(251−46, 251, 1556)-Net over F₄ — Constructive and digital

Digital (205, 251, 1556)-net over F₄, using

4¹ times duplication [i] based on digital (204, 250, 1556)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 28, 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 (176, 222, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 74, 513)-net over F₆₄, using
      - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(251−46, 251, 15121)-Net over F₄ — Digital

Digital (205, 251, 15121)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵¹, 15121, F₄, 46) (dual of [15121, 14870, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵¹, 16424, F₄, 46) (dual of [16424, 16173, 47]-code), using
  - 5 times code embedding in larger space [i] based on linear OA(4²⁴⁶, 16419, F₄, 46) (dual of [16419, 16173, 47]-code), using
    - constructionÂ X applied to C_e(45) âŠ‚ C_e(40) [i] based on
      1. linear OA(4²³⁹, 16384, F₄, 46) (dual of [16384, 16145, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
      2. linear OA(4²¹¹, 16384, F₄, 41) (dual of [16384, 16173, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(251−46, 251, large)-Net in Base 4 — Upper bound on s

There is no (205, 251, large)-net in base 4, because

44 times m-reduction [i] would yield (205, 207, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]