MinT - Best Known (27, 31, s)-Nets in Base 7

Best Known (27, 31, s)-Nets in Base 7

(27, 31, 5767605)-Net over F₇ — Constructive and digital

Digital (27, 31, 5767605)-net over F₇, using

net defined by OOA [i] based on linear OOA(7³¹, 5767605, F₇, 4, 4) (dual of [(5767605, 4), 23070389, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(7³¹, 5767605, F₇, 3, 4) (dual of [(5767605, 3), 17302784, 5]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(7⁵, 2801, F₇, 3, 2) (dual of [(2801, 3), 8398, 3]-NRT-code), using
      - appending 1 arbitrary column [i] based on linear OOA(7⁵, 2801, F₇, 2, 2) (dual of [(2801, 2), 5597, 3]-NRT-code), using
        appending kth column [i] based on linear OA(7⁵, 2801, F₇, 2) (dual of [2801, 2796, 3]-code), using
        Hamming code H(5,7) [i]
    2. linear OOA(7²⁶, 5764804, F₇, 3, 4) (dual of [(5764804, 3), 17294386, 5]-NRT-code), using
      - trace code [i] based on linear OOA(49¹³, 2882402, F₄₉, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(49¹³, 5764804, F₄₉, 4) (dual of [5764804, 5764791, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(49¹³, 5764805, F₄₉, 4) (dual of [5764805, 5764792, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁹, 5764801, F₄₉, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(27, 31, large)-Net over F₇ — Digital

Digital (27, 31, large)-net over F₇, using

7⁵ times duplication [i] based on digital (22, 26, large)-net over F₇, using
- net defined by OOA [i] based on linear OOA(7²⁶, large, F₇, 4, 4), using
  - appending kth column [i] based on linear OOA(7²⁶, large, F₇, 3, 4), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁶, large, F₇, 4) (dual of [large, large−26, 5]-code), using
      - trace code [i] based on linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using
        an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(27, 31, large)-Net in Base 7 — Upper bound on s

There is no (27, 31, large)-net in base 7, because

2 times m-reduction [i] would yield (27, 29, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (27, 31, s)-Nets in Base 7

(27, 31, 5767605)-Net over F7 — Constructive and digital

(27, 31, large)-Net over F7 — Digital

(27, 31, large)-Net in Base 7 — Upper bound on s

(27, 31, 5767605)-Net over F₇ — Constructive and digital

(27, 31, large)-Net over F₇ — Digital