Best Known (64, 64+26, s)-Nets in Base 8
(64, 64+26, 416)-Net over F8 — Constructive and digital
Digital (64, 90, 416)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (19, 32, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- digital (32, 58, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 29, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64 (see above)
- trace code for nets [i] based on digital (3, 29, 104)-net over F64, using
- digital (19, 32, 208)-net over F8, using
(64, 64+26, 576)-Net in Base 8 — Constructive
(64, 90, 576)-net in base 8, using
- 6 times m-reduction [i] based on (64, 96, 576)-net in base 8, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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, 42, 288)-net over F128, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
(64, 64+26, 3113)-Net over F8 — Digital
Digital (64, 90, 3113)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(890, 3113, F8, 26) (dual of [3113, 3023, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(890, 4101, F8, 26) (dual of [4101, 4011, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(889, 4100, F8, 26) (dual of [4100, 4011, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(889, 4096, F8, 26) (dual of [4096, 4007, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(885, 4096, F8, 25) (dual of [4096, 4011, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(889, 4100, F8, 26) (dual of [4100, 4011, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(890, 4101, F8, 26) (dual of [4101, 4011, 27]-code), using
(64, 64+26, 1446995)-Net in Base 8 — Upper bound on s
There is no (64, 90, 1446996)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1897 143393 787611 226311 989356 356108 729684 304208 383302 799842 130134 230907 238473 496692 > 890 [i]