MinT - Best Known (142−10, 142, s)-Nets in Base 4

Best Known (142−10, 142, s)-Nets in Base 4

(142−10, 142, 6841957)-Net over F₄ — Constructive and digital

Digital (132, 142, 6841957)-net over F₄, using

4¹ times duplication [i] based on digital (131, 141, 6841957)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (24, 29, 131077)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4²⁹, 131077, F₄, 5, 5) (dual of [(131077, 5), 655356, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(4²⁹, 262155, F₄, 5) (dual of [262155, 262126, 6]-code), using
        constructionÂ X4 applied to C_e(4) âŠ‚ C_e(2) [i] based on
        linear OA(4²⁸, 262144, F₄, 5) (dual of [262144, 262116, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(4¹⁹, 262144, F₄, 3) (dual of [262144, 262125, 4]-code or 262144-cap in PG(18,4)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(4¹⁰, 11, F₄, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
        dual of repetition code with length 11 [i]
        linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. digital (102, 112, 6710880)-net over F₄, using
    - trace code for nets [i] based on digital (18, 28, 1677720)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256²⁸, 1677720, F₂₅₆, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(256²⁸, 8388600, F₂₅₆, 10) (dual of [8388600, 8388572, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁸, large, F₂₅₆, 10) (dual of [large, large−28, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(142−10, 142, large)-Net over F₄ — Digital

Digital (132, 142, large)-net over F₄, using

t-expansion [i] based on digital (130, 142, large)-net over F₄, using
- 3 times m-reduction [i] based on digital (130, 145, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁵, large, F₄, 15) (dual of [large, large−145, 16]-code), using
    - strength reduction [i] based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(142−10, 142, large)-Net in Base 4 — Upper bound on s

There is no (132, 142, large)-net in base 4, because

8 times m-reduction [i] would yield (132, 134, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (142−10, 142, s)-Nets in Base 4

(142−10, 142, 6841957)-Net over F4 — Constructive and digital

(142−10, 142, large)-Net over F4 — Digital

(142−10, 142, large)-Net in Base 4 — Upper bound on s

(142−10, 142, 6841957)-Net over F₄ — Constructive and digital

(142−10, 142, large)-Net over F₄ — Digital