MinT - Best Known (152, 152+24, s)-Nets in Base 4

Best Known (152, 152+24, s)-Nets in Base 4

(152, 152+24, 21855)-Net over F₄ — Constructive and digital

Digital (152, 176, 21855)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (138, 162, 21845)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁶², 21845, F₄, 24, 24) (dual of [(21845, 24), 524118, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(4¹⁶², 262140, F₄, 24) (dual of [262140, 261978, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶², 262144, F₄, 24) (dual of [262144, 261982, 25]-code), using
        1 times truncation [i] based on linear OA(4¹⁶³, 262145, F₄, 25) (dual of [262145, 261982, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(152, 152+24, 185697)-Net over F₄ — Digital

Digital (152, 176, 185697)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁶, 185697, F₄, 24) (dual of [185697, 185521, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷⁶, 262209, F₄, 24) (dual of [262209, 262033, 25]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(4¹⁶³, 262145, F₄, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(4¹⁰⁹, 262145, F₄, 17) (dual of [262145, 262036, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    3. linear OA(4¹³, 64, F₄, 6) (dual of [64, 51, 7]-code), using
      - an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(152, 152+24, large)-Net in Base 4 — Upper bound on s

There is no (152, 176, large)-net in base 4, because

22 times m-reduction [i] would yield (152, 154, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]