Best Known (60, 60+23, s)-Nets in Base 7
(60, 60+23, 300)-Net over F7 — Constructive and digital
Digital (60, 83, 300)-net over F7, using
- 71 times duplication [i] based on digital (59, 82, 300)-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 (23, 46, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 50)-net over F49, using
- digital (7, 14, 100)-net over F7, using
- generalized (u, u+v)-construction [i] based on
(60, 60+23, 2506)-Net over F7 — Digital
Digital (60, 83, 2506)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 2506, F7, 23) (dual of [2506, 2423, 24]-code), using
- 95 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 57 times 0) [i] based on linear OA(777, 2405, F7, 23) (dual of [2405, 2328, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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]
- 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(22) ⊂ Ce(21) [i] based on
- 95 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 57 times 0) [i] based on linear OA(777, 2405, F7, 23) (dual of [2405, 2328, 24]-code), using
(60, 60+23, 1631859)-Net in Base 7 — Upper bound on s
There is no (60, 83, 1631860)-net in base 7, because
- 1 times m-reduction [i] would yield (60, 82, 1631860)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1986 277437 480330 679841 280278 366565 496335 968587 806785 635960 546651 995553 > 782 [i]