Best Known (87, 87+19, s)-Nets in Base 8
(87, 87+19, 29136)-Net over F8 — Constructive and digital
Digital (87, 106, 29136)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- a shift-net [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (78, 97, 29127)-net over F8, using
- net defined by OOA [i] based on linear OOA(897, 29127, F8, 19, 19) (dual of [(29127, 19), 553316, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(897, 262144, F8, 19) (dual of [262144, 262047, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(897, 262144, F8, 19) (dual of [262144, 262047, 20]-code), using
- net defined by OOA [i] based on linear OOA(897, 29127, F8, 19, 19) (dual of [(29127, 19), 553316, 20]-NRT-code), using
- digital (0, 9, 9)-net over F8, using
(87, 87+19, 262185)-Net over F8 — Digital
Digital (87, 106, 262185)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8106, 262185, F8, 19) (dual of [262185, 262079, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8104, 262181, F8, 19) (dual of [262181, 262077, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(897, 262144, F8, 19) (dual of [262144, 262047, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(8104, 262183, F8, 18) (dual of [262183, 262079, 19]-code), using Gilbert–Varšamov bound and bm = 8104 > Vbs−1(k−1) = 85 433014 311321 246639 375757 339977 814828 139860 345653 082245 067183 870574 296017 643039 030139 004442 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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
- linear OA(8104, 262181, F8, 19) (dual of [262181, 262077, 20]-code), using
- construction X with Varšamov bound [i] based on
(87, 87+19, large)-Net in Base 8 — Upper bound on s
There is no (87, 106, large)-net in base 8, because
- 17 times m-reduction [i] would yield (87, 89, large)-net in base 8, but