MinT - Best Known (127, 144, s)-Nets in Base 4

Best Known (127, 144, s)-Nets in Base 4

(127, 144, 524302)-Net over F₄ — Constructive and digital

Digital (127, 144, 524302)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (116, 133, 524288)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹³³, 524288, F₄, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(4¹³³, 4194305, F₄, 17) (dual of [4194305, 4194172, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(127, 144, 2097180)-Net over F₄ — Digital

Digital (127, 144, 2097180)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁴⁴, 2097180, F₄, 2, 17) (dual of [(2097180, 2), 4194216, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹⁴⁴, 4194360, F₄, 17) (dual of [4194360, 4194216, 18]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁴³, 4194359, F₄, 17) (dual of [4194359, 4194216, 18]-code), using
    - constructionÂ X applied to C([0,8]) âŠ‚ C([0,5]) [i] based on
      1. linear OA(4¹³³, 4194305, F₄, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      2. linear OA(4⁸⁹, 4194305, F₄, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
      3. linear OA(4¹⁰, 54, F₄, 5) (dual of [54, 44, 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]

(127, 144, large)-Net in Base 4 — Upper bound on s

There is no (127, 144, large)-net in base 4, because

15 times m-reduction [i] would yield (127, 129, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]