Best Known (57, 57+21, s)-Nets in Base 7
(57, 57+21, 302)-Net over F7 — Constructive and digital
Digital (57, 78, 302)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 14, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 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 (23, 44, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 22, 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, 22, 51)-net over F49, using
- digital (7, 14, 100)-net over F7, using
(57, 57+21, 2778)-Net over F7 — Digital
Digital (57, 78, 2778)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 2778, F7, 21) (dual of [2778, 2700, 22]-code), using
- 372 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 18 times 0, 1, 50 times 0, 1, 108 times 0, 1, 187 times 0) [i] based on linear OA(772, 2400, F7, 21) (dual of [2400, 2328, 22]-code), using
- 1 times truncation [i] based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- 372 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 18 times 0, 1, 50 times 0, 1, 108 times 0, 1, 187 times 0) [i] based on linear OA(772, 2400, F7, 21) (dual of [2400, 2328, 22]-code), using
(57, 57+21, 2427050)-Net in Base 7 — Upper bound on s
There is no (57, 78, 2427051)-net in base 7, because
- 1 times m-reduction [i] would yield (57, 77, 2427051)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 118181 508402 678756 658378 648858 377474 296183 926426 616005 708147 767709 > 777 [i]