Best Known (42, 68, s)-Nets in Base 7
(42, 68, 116)-Net over F7 — Constructive and digital
Digital (42, 68, 116)-net over F7, using
- trace code for nets [i] based on digital (8, 34, 58)-net over F49, using
- net from sequence [i] based on digital (8, 57)-sequence over F49, using
(42, 68, 358)-Net over F7 — Digital
Digital (42, 68, 358)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(768, 358, F7, 26) (dual of [358, 290, 27]-code), using
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0) [i] based on linear OA(767, 348, F7, 26) (dual of [348, 281, 27]-code), using
- construction XX applied to C1 = C([341,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([341,24]) [i] based on
- linear OA(764, 342, F7, 25) (dual of [342, 278, 26]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(764, 342, F7, 25) (dual of [342, 278, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(767, 342, F7, 26) (dual of [342, 275, 27]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(761, 342, F7, 24) (dual of [342, 281, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([341,24]) [i] based on
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0) [i] based on linear OA(767, 348, F7, 26) (dual of [348, 281, 27]-code), using
(42, 68, 24867)-Net in Base 7 — Upper bound on s
There is no (42, 68, 24868)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 2930 020700 052234 606243 418980 402808 300821 460289 501792 708857 > 768 [i]