MinT - Best Known (78, 78+11, s)-Nets in Base 7

Best Known (78, 78+11, s)-Nets in Base 7

(78, 78+11, 2305941)-Net over F₇ — Constructive and digital

Digital (78, 89, 2305941)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 21)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁷, 21, F₇, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [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 (71, 82, 2305920)-net over F₇, using
  - trace code for nets [i] based on digital (30, 41, 1152960)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁴¹, 1152960, F₄₉, 11, 11) (dual of [(1152960, 11), 12682519, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(49⁴¹, 5764801, F₄₉, 11) (dual of [5764801, 5764760, 12]-code), using
        an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(78, 78+11, large)-Net over F₇ — Digital

Digital (78, 89, large)-net over F₇, using

1 times m-reduction [i] based on digital (78, 90, large)-net over F₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁰, large, F₇, 12) (dual of [large, large−90, 13]-code), using
  - trace code [i] based on linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using
    - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(78, 78+11, large)-Net in Base 7 — Upper bound on s

There is no (78, 89, large)-net in base 7, because

9 times m-reduction [i] would yield (78, 80, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]