Best Known (72, 72+28, s)-Nets in Base 8
(72, 72+28, 514)-Net over F8 — Constructive and digital
Digital (72, 100, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 30, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 15, 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, 15, 80)-net over F64, using
- digital (42, 70, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 177)-net over F64, using
- digital (16, 30, 160)-net over F8, using
(72, 72+28, 576)-Net in Base 8 — Constructive
(72, 100, 576)-net in base 8, using
- 10 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
- 1 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
- 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
(72, 72+28, 4120)-Net over F8 — Digital
Digital (72, 100, 4120)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8100, 4120, F8, 28) (dual of [4120, 4020, 29]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 12 times 0) [i] based on linear OA(897, 4100, F8, 28) (dual of [4100, 4003, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(897, 4096, F8, 28) (dual of [4096, 3999, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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(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(27) ⊂ Ce(26) [i] based on
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 12 times 0) [i] based on linear OA(897, 4100, F8, 28) (dual of [4100, 4003, 29]-code), using
(72, 72+28, 2437816)-Net in Base 8 — Upper bound on s
There is no (72, 100, 2437817)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2 037045 948874 112941 579784 735141 488663 375636 538956 874823 975547 005690 736085 748975 551365 544664 > 8100 [i]