Best Known (57, 57+20, s)-Nets in Base 7
(57, 57+20, 308)-Net over F7 — Constructive and digital
Digital (57, 77, 308)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 5, 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 (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 (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 (see above)
- trace code for nets [i] based on digital (0, 10, 50)-net over F49, using
- digital (20, 40, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 50)-net over F49, using
- digital (0, 5, 8)-net over F7, using
(57, 57+20, 3526)-Net over F7 — Digital
Digital (57, 77, 3526)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(777, 3526, F7, 20) (dual of [3526, 3449, 21]-code), using
- 1113 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, 1, 337 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
- 1113 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, 1, 337 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
(57, 57+20, 2427050)-Net in Base 7 — Upper bound on s
There is no (57, 77, 2427051)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 118181 508402 678756 658378 648858 377474 296183 926426 616005 708147 767709 > 777 [i]