Best Known (102−26, 102, s)-Nets in Base 7
(102−26, 102, 688)-Net over F7 — Constructive and digital
Digital (76, 102, 688)-net over F7, using
- 8 times m-reduction [i] based on digital (76, 110, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 55, 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, 55, 344)-net over F49, using
(102−26, 102, 4806)-Net over F7 — Digital
Digital (76, 102, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7102, 4806, F7, 26) (dual of [4806, 4704, 27]-code), using
- trace code [i] based on linear OA(4951, 2403, F49, 26) (dual of [2403, 2352, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4951, 2401, F49, 26) (dual of [2401, 2350, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4949, 2401, F49, 25) (dual of [2401, 2352, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- trace code [i] based on linear OA(4951, 2403, F49, 26) (dual of [2403, 2352, 27]-code), using
(102−26, 102, 4036670)-Net in Base 7 — Upper bound on s
There is no (76, 102, 4036671)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 158 489673 605878 580951 001154 562659 443488 518963 651153 726988 045348 317300 836629 468079 715123 > 7102 [i]