MinT - Best Known (182−32, 182, s)-Nets in Base 4

Best Known (182−32, 182, s)-Nets in Base 4

(182−32, 182, 1539)-Net over F₄ — Constructive and digital

Digital (150, 182, 1539)-net over F₄, using

1 times m-reduction [i] based on digital (150, 183, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 61, 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]

(182−32, 182, 16440)-Net over F₄ — Digital

Digital (150, 182, 16440)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸², 16440, F₄, 32) (dual of [16440, 16258, 33]-code), using
- constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
  1. linear OA(4¹⁶⁹, 16385, F₄, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
  2. linear OA(4¹²⁷, 16385, F₄, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
  3. linear OA(4¹³, 55, F₄, 6) (dual of [55, 42, 7]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹³, 63, F₄, 6) (dual of [63, 50, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(182−32, 182, large)-Net in Base 4 — Upper bound on s

There is no (150, 182, large)-net in base 4, because

30 times m-reduction [i] would yield (150, 152, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]