MinT - Best Known (139−13, 139, s)-Nets in Base 4

Best Known (139−13, 139, s)-Nets in Base 4

(139−13, 139, 1403564)-Net over F₄ — Constructive and digital

Digital (126, 139, 1403564)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (24, 30, 5464)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4³⁰, 5464, F₄, 6, 6) (dual of [(5464, 6), 32754, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(4³⁰, 16392, F₄, 6) (dual of [16392, 16362, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³⁰, 16393, F₄, 6) (dual of [16393, 16363, 7]-code), using
        constructionÂ X4 applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(4²⁹, 16384, F₄, 6) (dual of [16384, 16355, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4²², 16384, F₄, 5) (dual of [16384, 16362, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(4⁸, 9, F₄, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
        dual of repetition code with length 9 [i]
        linear OA(4¹, 9, F₄, 1) (dual of [9, 8, 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 (96, 109, 1398100)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁰⁹, 1398100, F₄, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(4¹⁰⁹, 8388601, F₄, 13) (dual of [8388601, 8388492, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁰⁹, large, F₄, 13) (dual of [large, large−109, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(139−13, 139, large)-Net over F₄ — Digital

Digital (126, 139, large)-net over F₄, using

4⁴ times duplication [i] based on digital (122, 135, large)-net over F₄, using
- t-expansion [i] based on digital (121, 135, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁵, large, F₄, 14) (dual of [large, large−135, 15]-code), using
    - 14 times code embedding in larger space [i] based on linear OA(4¹²¹, large, F₄, 14) (dual of [large, large−121, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(139−13, 139, large)-Net in Base 4 — Upper bound on s

There is no (126, 139, large)-net in base 4, because

11 times m-reduction [i] would yield (126, 128, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (139−13, 139, s)-Nets in Base 4

(139−13, 139, 1403564)-Net over F4 — Constructive and digital

(139−13, 139, large)-Net over F4 — Digital

(139−13, 139, large)-Net in Base 4 — Upper bound on s

(139−13, 139, 1403564)-Net over F₄ — Constructive and digital

(139−13, 139, large)-Net over F₄ — Digital