Best Known (28−10, 28, s)-Nets in Base 7
(28−10, 28, 128)-Net over F7 — Constructive and digital
Digital (18, 28, 128)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 8, 28)-net over F7, using
- digital (10, 20, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 50)-net over F49, using
(28−10, 28, 370)-Net over F7 — Digital
Digital (18, 28, 370)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(728, 370, F7, 10) (dual of [370, 342, 11]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(725, 348, F7, 10) (dual of [348, 323, 11]-code), using
- construction XX applied to C1 = C([341,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([341,8]) [i] based on
- linear OA(722, 342, F7, 9) (dual of [342, 320, 10]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(722, 342, F7, 9) (dual of [342, 320, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(725, 342, F7, 10) (dual of [342, 317, 11]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(719, 342, F7, 8) (dual of [342, 323, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [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([341,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([341,8]) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(725, 348, F7, 10) (dual of [348, 323, 11]-code), using
(28−10, 28, 23452)-Net in Base 7 — Upper bound on s
There is no (18, 28, 23453)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 460075 736050 229977 975087 > 728 [i]