Best Known (48−8, 48, s)-Nets in Base 7
(48−8, 48, 205898)-Net over F7 — Constructive and digital
Digital (40, 48, 205898)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 13)-net over F7, using
- 8 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (35, 43, 205885)-net over F7, using
- net defined by OOA [i] based on linear OOA(743, 205885, F7, 8, 8) (dual of [(205885, 8), 1647037, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(743, 823540, F7, 8) (dual of [823540, 823497, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(743, 823540, F7, 8) (dual of [823540, 823497, 9]-code), using
- net defined by OOA [i] based on linear OOA(743, 205885, F7, 8, 8) (dual of [(205885, 8), 1647037, 9]-NRT-code), using
- digital (1, 5, 13)-net over F7, using
(48−8, 48, 823570)-Net over F7 — Digital
Digital (40, 48, 823570)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(748, 823570, F7, 8) (dual of [823570, 823522, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(715, 823543, F7, 3) (dual of [823543, 823528, 4]-code or 823543-cap in PG(14,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(74, 26, F7, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
(48−8, 48, large)-Net in Base 7 — Upper bound on s
There is no (40, 48, large)-net in base 7, because
- 6 times m-reduction [i] would yield (40, 42, large)-net in base 7, but