Best Known (105−31, 105, s)-Nets in Base 7
(105−31, 105, 688)-Net over F7 — Constructive and digital
Digital (74, 105, 688)-net over F7, using
- 1 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
(105−31, 105, 2070)-Net over F7 — Digital
Digital (74, 105, 2070)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7105, 2070, F7, 31) (dual of [2070, 1965, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(7105, 2401, F7, 31) (dual of [2401, 2296, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(7105, 2401, F7, 31) (dual of [2401, 2296, 32]-code), using
(105−31, 105, 774384)-Net in Base 7 — Upper bound on s
There is no (74, 105, 774385)-net in base 7, because
- 1 times m-reduction [i] would yield (74, 104, 774385)-net in base 7, but
- 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]