Best Known (37, 37+5, s)-Nets in Base 7
(37, 37+5, 8388602)-Net over F7 — Constructive and digital
Digital (37, 42, 8388602)-net over F7, using
- trace code for nets [i] based on digital (16, 21, 4194301)-net over F49, using
- net defined by OOA [i] based on linear OOA(4921, 4194301, F49, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4921, large, F49, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 28247525 | 4910−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(4921, large, F49, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(4921, 4194301, F49, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
(37, 37+5, large)-Net over F7 — Digital
Digital (37, 42, large)-net over F7, using
- t-expansion [i] based on digital (36, 42, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(742, large, F7, 6) (dual of [large, large−42, 7]-code), using
- trace code [i] based on linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- trace code [i] based on linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(742, large, F7, 6) (dual of [large, large−42, 7]-code), using
(37, 37+5, large)-Net in Base 7 — Upper bound on s
There is no (37, 42, large)-net in base 7, because
- 3 times m-reduction [i] would yield (37, 39, large)-net in base 7, but