Best Known (104−30, 104, s)-Nets in Base 7
(104−30, 104, 688)-Net over F7 — Constructive and digital
Digital (74, 104, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (74, 106, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 53, 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, 53, 344)-net over F49, using
(104−30, 104, 2402)-Net over F7 — Digital
Digital (74, 104, 2402)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 2402, F7, 30) (dual of [2402, 2298, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 2412, F7, 30) (dual of [2412, 2308, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(7101, 2401, F7, 30) (dual of [2401, 2300, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(793, 2401, F7, 27) (dual of [2401, 2308, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(73, 11, F7, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 2412, F7, 30) (dual of [2412, 2308, 31]-code), using
(104−30, 104, 774384)-Net in Base 7 — Upper bound on s
There is no (74, 104, 774385)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 7766 101127 512122 484004 917072 099228 732881 919400 479171 626585 118370 795587 374488 427270 042295 > 7104 [i]