Best Known (137−21, 137, s)-Nets in Base 8
(137−21, 137, 209724)-Net over F8 — Constructive and digital
Digital (116, 137, 209724)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (106, 127, 209715)-net over F8, using
- net defined by OOA [i] based on linear OOA(8127, 209715, F8, 21, 21) (dual of [(209715, 21), 4403888, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8127, 2097151, F8, 21) (dual of [2097151, 2097024, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8127, 2097151, F8, 21) (dual of [2097151, 2097024, 22]-code), using
- net defined by OOA [i] based on linear OOA(8127, 209715, F8, 21, 21) (dual of [(209715, 21), 4403888, 22]-NRT-code), using
- digital (0, 10, 9)-net over F8, using
(137−21, 137, 2097204)-Net over F8 — Digital
Digital (116, 137, 2097204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8137, 2097204, F8, 21) (dual of [2097204, 2097067, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(885, 2097152, F8, 14) (dual of [2097152, 2097067, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(810, 52, F8, 6) (dual of [52, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
(137−21, 137, large)-Net in Base 8 — Upper bound on s
There is no (116, 137, large)-net in base 8, because
- 19 times m-reduction [i] would yield (116, 118, large)-net in base 8, but