Best Known (16, 26, s)-Nets in Base 7
(16, 26, 113)-Net over F7 — Constructive and digital
Digital (16, 26, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 13)-net over F7, using
- 7 times m-reduction [i] based on digital (1, 13, 13)-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
- digital (1, 6, 13)-net over F7, using
(16, 26, 270)-Net over F7 — Digital
Digital (16, 26, 270)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(726, 270, F7, 10) (dual of [270, 244, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(726, 350, F7, 10) (dual of [350, 324, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(725, 343, F7, 10) (dual of [343, 318, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(719, 343, F7, 8) (dual of [343, 324, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(726, 350, F7, 10) (dual of [350, 324, 11]-code), using
(16, 26, 10766)-Net in Base 7 — Upper bound on s
There is no (16, 26, 10767)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 9389 011332 696294 645955 > 726 [i]