Best Known (82−22, 82, s)-Nets in Base 7
(82−22, 82, 302)-Net over F7 — Constructive and digital
Digital (60, 82, 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 (11, 22, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 50)-net over F49, using
- digital (24, 46, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 23, 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, 23, 51)-net over F49, using
- digital (7, 14, 100)-net over F7, using
(82−22, 82, 2919)-Net over F7 — Digital
Digital (60, 82, 2919)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(782, 2919, F7, 22) (dual of [2919, 2837, 23]-code), using
- 509 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 5 times 0, 1, 16 times 0, 1, 39 times 0, 1, 84 times 0, 1, 146 times 0, 1, 211 times 0) [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]
- 509 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 5 times 0, 1, 16 times 0, 1, 39 times 0, 1, 84 times 0, 1, 146 times 0, 1, 211 times 0) [i] based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
(82−22, 82, 1631859)-Net in Base 7 — Upper bound on s
There is no (60, 82, 1631860)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1986 277437 480330 679841 280278 366565 496335 968587 806785 635960 546651 995553 > 782 [i]