MinT - Best Known (39, 39+7, s)-Nets in Base 7

Best Known (39, 39+7, s)-Nets in Base 7

(39, 39+7, 274563)-Net over F₇ — Constructive and digital

Digital (39, 46, 274563)-net over F₇, using

net defined by OOA [i] based on linear OOA(7⁴⁶, 274563, F₇, 9, 7) (dual of [(274563, 9), 2471021, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7⁴⁶, 274564, F₇, 3, 7) (dual of [(274564, 3), 823646, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(7⁴, 50, F₇, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(7⁴, 50, F₇, 2, 3) (dual of [(50, 2), 96, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁴, 50, F₇, 3) (dual of [50, 46, 4]-code or 50-cap in PG(3,7)), using
        ovoid in PG(3,Â 7) [i]
    2. linear OOA(7⁴², 274514, F₇, 3, 7) (dual of [(274514, 3), 823500, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(7⁴², 823542, F₇, 7) (dual of [823542, 823500, 8]-code), using
        the primitive narrow-sense BCH-code C(I) with length 823542Â = 7⁷âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(39, 39+7, 1503684)-Net over F₇ — Digital

Digital (39, 46, 1503684)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+7, large)-Net in Base 7 — Upper bound on s

There is no (39, 46, large)-net in base 7, because

5 times m-reduction [i] would yield (39, 41, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]