Best Known (56, 56+20, s)-Nets in Base 7
(56, 56+20, 304)-Net over F7 — Constructive and digital
Digital (56, 76, 304)-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 (12, 22, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 11, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
- trace code for nets [i] based on digital (1, 11, 51)-net over F49, using
- digital (22, 42, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 21, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49 (see above)
- trace code for nets [i] based on digital (1, 21, 51)-net over F49, using
- digital (6, 12, 100)-net over F7, using
(56, 56+20, 3187)-Net over F7 — Digital
Digital (56, 76, 3187)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(776, 3187, F7, 20) (dual of [3187, 3111, 21]-code), using
- 775 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0, 1, 71 times 0, 1, 147 times 0, 1, 229 times 0, 1, 292 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 775 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0, 1, 71 times 0, 1, 147 times 0, 1, 229 times 0, 1, 292 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
(56, 56+20, 1997877)-Net in Base 7 — Upper bound on s
There is no (56, 76, 1997878)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 16883 112791 313201 433728 138475 200386 831140 186685 297991 102402 148909 > 776 [i]