Best Known (69, 69+27, s)-Nets in Base 8
(69, 69+27, 514)-Net over F8 — Constructive and digital
Digital (69, 96, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (15, 28, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
- digital (41, 68, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 177)-net over F64, using
- digital (15, 28, 160)-net over F8, using
(69, 69+27, 576)-Net in Base 8 — Constructive
(69, 96, 576)-net in base 8, using
- 8 times m-reduction [i] based on (69, 104, 576)-net in base 8, using
- trace code for nets [i] based on (17, 52, 288)-net in base 64, using
- 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- trace code for nets [i] based on (17, 52, 288)-net in base 64, using
(69, 69+27, 3914)-Net over F8 — Digital
Digital (69, 96, 3914)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(896, 3914, F8, 27) (dual of [3914, 3818, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(896, 4108, F8, 27) (dual of [4108, 4012, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(893, 4096, F8, 27) (dual of [4096, 4003, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(881, 4096, F8, 23) (dual of [4096, 4015, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 10, F8, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(81, 2, F8, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(896, 4108, F8, 27) (dual of [4108, 4012, 28]-code), using
(69, 69+27, 3219666)-Net in Base 8 — Upper bound on s
There is no (69, 96, 3219667)-net in base 8, because
- 1 times m-reduction [i] would yield (69, 95, 3219667)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 62 165640 650965 856598 393728 720489 781574 812712 109654 764963 295986 433035 293701 376260 914952 > 895 [i]