MinT - Best Known (257−47, 257, s)-Nets in Base 4

Best Known (257−47, 257, s)-Nets in Base 4

(257−47, 257, 1561)-Net over F₄ — Constructive and digital

Digital (210, 257, 1561)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 32, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (178, 225, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 75, 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]

(257−47, 257, 15599)-Net over F₄ — Digital

Digital (210, 257, 15599)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁷, 15599, F₄, 47) (dual of [15599, 15342, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁷, 16430, F₄, 47) (dual of [16430, 16173, 48]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²⁵⁶, 16429, F₄, 47) (dual of [16429, 16173, 48]-code), using
    - constructionÂ X applied to C_e(46) âŠ‚ C_e(40) [i] based on
      1. linear OA(4²⁴⁶, 16384, F₄, 47) (dual of [16384, 16138, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [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(4¹⁰, 45, F₄, 5) (dual of [45, 35, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(257−47, 257, large)-Net in Base 4 — Upper bound on s

There is no (210, 257, large)-net in base 4, because

45 times m-reduction [i] would yield (210, 212, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]