MinT - Best Known (42−9, 42, s)-Nets in Base 7

Best Known (42−9, 42, s)-Nets in Base 7

(42−9, 42, 4223)-Net over F₇ — Constructive and digital

Digital (33, 42, 4223)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 21)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁶, 21, F₇, 4, 4) (dual of [(21, 4), 78, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(7⁶, 21, F₇, 3, 4) (dual of [(21, 3), 57, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(7⁶, 42, F₇, 4) (dual of [42, 36, 5]-code), using
        1 times truncation [i] based on linear OA(7⁷, 43, F₇, 5) (dual of [43, 36, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (27, 36, 4202)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7³⁶, 4202, F₇, 9, 9) (dual of [(4202, 9), 37782, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(7³⁶, 16809, F₇, 9) (dual of [16809, 16773, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(7³⁶, 16812, F₇, 9) (dual of [16812, 16776, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(7³⁶, 16807, F₇, 9) (dual of [16807, 16771, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(7³¹, 16807, F₇, 8) (dual of [16807, 16776, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(7⁰, 5, F₇, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(42−9, 42, 17155)-Net over F₇ — Digital

Digital (33, 42, 17155)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−9, 42, large)-Net in Base 7 — Upper bound on s

There is no (33, 42, large)-net in base 7, because

7 times m-reduction [i] would yield (33, 35, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (42−9, 42, s)-Nets in Base 7

(42−9, 42, 4223)-Net over F7 — Constructive and digital

(42−9, 42, 17155)-Net over F7 — Digital

(42−9, 42, large)-Net in Base 7 — Upper bound on s

(42−9, 42, 4223)-Net over F₇ — Constructive and digital

(42−9, 42, 17155)-Net over F₇ — Digital