Best Known (68, 93, s)-Nets in Base 7
(68, 93, 688)-Net over F7 — Constructive and digital
Digital (68, 93, 688)-net over F7, using
- 1 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, 93, 3090)-Net over F7 — Digital
Digital (68, 93, 3090)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(793, 3090, F7, 25) (dual of [3090, 2997, 26]-code), using
- 677 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 72 times 0, 1, 133 times 0, 1, 191 times 0, 1, 229 times 0) [i] based on linear OA(785, 2405, F7, 25) (dual of [2405, 2320, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(785, 2401, F7, 25) (dual of [2401, 2316, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 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(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(24) ⊂ Ce(23) [i] based on
- 677 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 72 times 0, 1, 133 times 0, 1, 191 times 0, 1, 229 times 0) [i] based on linear OA(785, 2405, F7, 25) (dual of [2405, 2320, 26]-code), using
(68, 93, 2656402)-Net in Base 7 — Upper bound on s
There is no (68, 93, 2656403)-net in base 7, because
- 1 times m-reduction [i] would yield (68, 92, 2656403)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 561074 200114 649708 997994 066322 064299 537751 888296 780370 004990 852121 934261 719257 > 792 [i]