Best Known (49, 49+30, s)-Nets in Base 8
(49, 49+30, 354)-Net over F8 — Constructive and digital
Digital (49, 79, 354)-net over F8, using
- 5 times m-reduction [i] based on digital (49, 84, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 177)-net over F64, using
(49, 49+30, 384)-Net in Base 8 — Constructive
(49, 79, 384)-net in base 8, using
- 1 times m-reduction [i] based on (49, 80, 384)-net in base 8, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
- 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
(49, 49+30, 514)-Net over F8 — Digital
Digital (49, 79, 514)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(879, 514, F8, 30) (dual of [514, 435, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(879, 515, F8, 30) (dual of [515, 436, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(879, 512, F8, 30) (dual of [512, 433, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(876, 512, F8, 29) (dual of [512, 436, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 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(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(879, 515, F8, 30) (dual of [515, 436, 31]-code), using
(49, 49+30, 52343)-Net in Base 8 — Upper bound on s
There is no (49, 79, 52344)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 220857 575738 004099 304980 401381 612585 365130 337254 725985 794974 709084 935930 > 879 [i]