Best Known (149−43, 149, s)-Nets in Base 8
(149−43, 149, 1026)-Net over F8 — Constructive and digital
Digital (106, 149, 1026)-net over F8, using
- 7 times m-reduction [i] based on digital (106, 156, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 78, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 78, 513)-net over F64, using
(149−43, 149, 4100)-Net over F8 — Digital
Digital (106, 149, 4100)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8149, 4100, F8, 43) (dual of [4100, 3951, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- linear OA(8149, 4096, F8, 43) (dual of [4096, 3947, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(8145, 4096, F8, 42) (dual of [4096, 3951, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
(149−43, 149, 2870957)-Net in Base 8 — Upper bound on s
There is no (106, 149, 2870958)-net in base 8, because
- 1 times m-reduction [i] would yield (106, 148, 2870958)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 45 427605 695345 870512 876859 306558 453226 920795 620691 587383 161895 953745 582010 676654 925908 594243 801511 450233 570313 053290 147479 697816 994114 > 8148 [i]