Best Known (42−4, 42, s)-Nets in Base 7
(42−4, 42, large)-Net over F7 — Constructive and digital
Digital (38, 42, large)-net over F7, using
- 1 times m-reduction [i] based on digital (38, 43, large)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (7, 9, 6725601)-net over F7, using
- digital (29, 34, 5764804)-net over F7, using
- trace code for nets [i] based on digital (12, 17, 2882402)-net over F49, using
- net defined by OOA [i] based on linear OOA(4917, 2882402, F49, 5, 5) (dual of [(2882402, 5), 14411993, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(4917, 5764805, F49, 5) (dual of [5764805, 5764788, 6]-code), using
- net defined by OOA [i] based on linear OOA(4917, 2882402, F49, 5, 5) (dual of [(2882402, 5), 14411993, 6]-NRT-code), using
- trace code for nets [i] based on digital (12, 17, 2882402)-net over F49, using
- (u, u+v)-construction [i] based on
(42−4, 42, large)-Net in Base 7 — Upper bound on s
There is no (38, 42, large)-net in base 7, because
- 2 times m-reduction [i] would yield (38, 40, large)-net in base 7, but