Best Known (77−22, 77, s)-Nets in Base 8
(77−22, 77, 389)-Net over F8 — Constructive and digital
Digital (55, 77, 389)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 19, 35)-net over F8, using
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 7, N(F) = 34, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 7 and N(F) ≥ 34, using a function field by Sémirat [i]
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- digital (36, 58, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 29, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 29, 177)-net over F64, using
- digital (8, 19, 35)-net over F8, using
(77−22, 77, 576)-Net in Base 8 — Constructive
(55, 77, 576)-net in base 8, using
- 3 times m-reduction [i] based on (55, 80, 576)-net in base 8, using
- trace code for nets [i] based on (15, 40, 288)-net in base 64, using
- 2 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- 2 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- trace code for nets [i] based on (15, 40, 288)-net in base 64, using
(77−22, 77, 3194)-Net over F8 — Digital
Digital (55, 77, 3194)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(877, 3194, F8, 22) (dual of [3194, 3117, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(877, 4096, F8, 22) (dual of [4096, 4019, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(877, 4096, F8, 22) (dual of [4096, 4019, 23]-code), using
(77−22, 77, 1470767)-Net in Base 8 — Upper bound on s
There is no (55, 77, 1470768)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 3450 880823 455682 235744 191941 507366 561465 920793 705556 524969 154196 195439 > 877 [i]