Best Known (18, 18+11, s)-Nets in Base 7
(18, 18+11, 121)-Net over F7 — Constructive and digital
Digital (18, 29, 121)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F7, using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- digital (11, 22, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 50)-net over F49, using
- digital (2, 7, 21)-net over F7, using
(18, 18+11, 290)-Net over F7 — Digital
Digital (18, 29, 290)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(729, 290, F7, 11) (dual of [290, 261, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(729, 350, F7, 11) (dual of [350, 321, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(728, 343, F7, 11) (dual of [343, 315, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(722, 343, F7, 9) (dual of [343, 321, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(729, 350, F7, 11) (dual of [350, 321, 12]-code), using
(18, 18+11, 23452)-Net in Base 7 — Upper bound on s
There is no (18, 29, 23453)-net in base 7, because
- 1 times m-reduction [i] would yield (18, 28, 23453)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 460075 736050 229977 975087 > 728 [i]