Best Known (94−24, 94, s)-Nets in Base 7
(94−24, 94, 688)-Net over F7 — Constructive and digital
Digital (70, 94, 688)-net over F7, using
- 4 times m-reduction [i] based on digital (70, 98, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 49, 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, 49, 344)-net over F49, using
(94−24, 94, 4806)-Net over F7 — Digital
Digital (70, 94, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(794, 4806, F7, 24) (dual of [4806, 4712, 25]-code), using
- trace code [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(4947, 2401, F49, 24) (dual of [2401, 2354, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- trace code [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
(94−24, 94, 3674040)-Net in Base 7 — Upper bound on s
There is no (70, 94, 3674041)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 27 492653 623358 967255 564867 227633 590079 244281 574130 233806 111472 235643 051015 562721 > 794 [i]