Best Known (67, 90, s)-Nets in Base 7
(67, 90, 688)-Net over F7 — Constructive and digital
Digital (67, 90, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (67, 92, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 46, 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, 46, 344)-net over F49, using
(67, 90, 4806)-Net over F7 — Digital
Digital (67, 90, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(790, 4806, F7, 23) (dual of [4806, 4716, 24]-code), using
- trace code [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(4943, 2401, F49, 22) (dual of [2401, 2358, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
(67, 90, 5629551)-Net in Base 7 — Upper bound on s
There is no (67, 90, 5629552)-net in base 7, because
- 1 times m-reduction [i] would yield (67, 89, 5629552)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1635 784243 263281 337506 026291 878532 902737 324808 401390 827010 636959 403432 192257 > 789 [i]