Best Known (49, 49+18, s)-Nets in Base 7
(49, 49+18, 300)-Net over F7 — Constructive and digital
Digital (49, 67, 300)-net over F7, using
- 1 times m-reduction [i] based on digital (49, 68, 300)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 12, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 50)-net over F49, using
- digital (9, 18, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 9, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 9, 50)-net over F49, using
- digital (19, 38, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 19, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 19, 50)-net over F49, using
- digital (6, 12, 100)-net over F7, using
- generalized (u, u+v)-construction [i] based on
(49, 49+18, 2670)-Net over F7 — Digital
Digital (49, 67, 2670)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(767, 2670, F7, 18) (dual of [2670, 2603, 19]-code), using
- 259 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 69 times 0, 1, 145 times 0) [i] based on linear OA(761, 2405, F7, 18) (dual of [2405, 2344, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 259 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 69 times 0, 1, 145 times 0) [i] based on linear OA(761, 2405, F7, 18) (dual of [2405, 2344, 19]-code), using
(49, 49+18, 1351710)-Net in Base 7 — Upper bound on s
There is no (49, 67, 1351711)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 418 377920 243679 540420 517455 439358 467593 097275 395949 342859 > 767 [i]