MinT - Best Known (257−32, 257, s)-Nets in Base 4

Best Known (257−32, 257, s)-Nets in Base 4

(257−32, 257, 65545)-Net over F₄ — Constructive and digital

Digital (225, 257, 65545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (208, 240, 65536)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²⁴⁰, 65536, F₄, 32, 32) (dual of [(65536, 32), 2096912, 33]-NRT-code), using
    - OA 16-folding and stacking [i] based on linear OA(4²⁴⁰, 1048576, F₄, 32) (dual of [1048576, 1048336, 33]-code), using
      - 1 times truncation [i] based on linear OA(4²⁴¹, 1048577, F₄, 33) (dual of [1048577, 1048336, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(257−32, 257, 551096)-Net over F₄ — Digital

Digital (225, 257, 551096)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁷, 551096, F₄, 32) (dual of [551096, 550839, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁷, 1048653, F₄, 32) (dual of [1048653, 1048396, 33]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(4²⁴¹, 1048577, F₄, 33) (dual of [1048577, 1048336, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹⁸¹, 1048577, F₄, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(4¹⁶, 76, F₄, 6) (dual of [76, 60, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶, 85, F₄, 6) (dual of [85, 69, 7]-code), using
        the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(257−32, 257, large)-Net in Base 4 — Upper bound on s

There is no (225, 257, large)-net in base 4, because

30 times m-reduction [i] would yield (225, 227, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]