Best Known (145−42, 145, s)-Nets in Base 8
(145−42, 145, 1026)-Net over F8 — Constructive and digital
Digital (103, 145, 1026)-net over F8, using
- 5 times m-reduction [i] based on digital (103, 150, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
(145−42, 145, 3992)-Net over F8 — Digital
Digital (103, 145, 3992)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8145, 3992, F8, 42) (dual of [3992, 3847, 43]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(8145, 4096, F8, 42) (dual of [4096, 3951, 43]-code), using
(145−42, 145, 2133109)-Net in Base 8 — Upper bound on s
There is no (103, 145, 2133110)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 88725 569505 477105 065556 764912 711584 398495 529065 628648 492529 506939 467628 013044 742732 380631 862638 139618 172058 916574 304445 263497 988477 > 8145 [i]