Best Known (67−25, 67, s)-Nets in Base 7
(67−25, 67, 118)-Net over F7 — Constructive and digital
Digital (42, 67, 118)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 18)-net over F7, using
- 1 times m-reduction [i] based on digital (5, 18, 18)-net over F7, using
- digital (25, 50, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 25, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 25, 50)-net over F49, using
- digital (5, 17, 18)-net over F7, using
(67−25, 67, 387)-Net over F7 — Digital
Digital (42, 67, 387)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(767, 387, F7, 25) (dual of [387, 320, 26]-code), using
- 36 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 10 times 0, 1, 22 times 0) [i] based on linear OA(764, 348, F7, 25) (dual of [348, 284, 26]-code), using
- construction XX applied to C1 = C([341,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([341,23]) [i] based on
- linear OA(761, 342, F7, 24) (dual of [342, 281, 25]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [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(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(758, 342, F7, 23) (dual of [342, 284, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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([341,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([341,23]) [i] based on
- 36 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 10 times 0, 1, 22 times 0) [i] based on linear OA(764, 348, F7, 25) (dual of [348, 284, 26]-code), using
(67−25, 67, 39189)-Net in Base 7 — Upper bound on s
There is no (42, 67, 39190)-net in base 7, because
- 1 times m-reduction [i] would yield (42, 66, 39190)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 59 783096 002656 478624 410180 562375 728187 478751 073567 271353 > 766 [i]