Best Known (68, 68+24, s)-Nets in Base 7
(68, 68+24, 688)-Net over F7 — Constructive and digital
Digital (68, 92, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (68, 94, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 47, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 47, 344)-net over F49, using
(68, 68+24, 3785)-Net over F7 — Digital
Digital (68, 92, 3785)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(792, 3785, F7, 24) (dual of [3785, 3693, 25]-code), using
- 1369 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 17 times 0, 1, 42 times 0, 1, 91 times 0, 1, 157 times 0, 1, 213 times 0, 1, 251 times 0, 1, 278 times 0, 1, 304 times 0) [i] based on linear OA(781, 2405, F7, 24) (dual of [2405, 2324, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(781, 2401, F7, 24) (dual of [2401, 2320, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(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(23) ⊂ Ce(22) [i] based on
- 1369 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 17 times 0, 1, 42 times 0, 1, 91 times 0, 1, 157 times 0, 1, 213 times 0, 1, 251 times 0, 1, 278 times 0, 1, 304 times 0) [i] based on linear OA(781, 2405, F7, 24) (dual of [2405, 2324, 25]-code), using
(68, 68+24, 2656402)-Net in Base 7 — Upper bound on s
There is no (68, 92, 2656403)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 561074 200114 649708 997994 066322 064299 537751 888296 780370 004990 852121 934261 719257 > 792 [i]