Best Known (35−13, 35, s)-Nets in Base 7
(35−13, 35, 116)-Net over F7 — Constructive and digital
Digital (22, 35, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 16)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (0, 6, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7 (see above)
- digital (0, 3, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 50)-net over F49, using
- digital (3, 9, 16)-net over F7, using
(35−13, 35, 329)-Net over F7 — Digital
Digital (22, 35, 329)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(735, 329, F7, 13) (dual of [329, 294, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(735, 350, F7, 13) (dual of [350, 315, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(734, 343, F7, 13) (dual of [343, 309, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(735, 350, F7, 13) (dual of [350, 315, 14]-code), using
(35−13, 35, 30683)-Net in Base 7 — Upper bound on s
There is no (22, 35, 30684)-net in base 7, because
- 1 times m-reduction [i] would yield (22, 34, 30684)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 54118 288133 774129 431813 733065 > 734 [i]