Best Known (52, 52+9, s)-Nets in Base 8
(52, 52+9, 2097159)-Net over F8 — Constructive and digital
Digital (52, 61, 2097159)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (48, 57, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(857, 2097150, F8, 9, 9) (dual of [(2097150, 9), 18874293, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(857, 8388601, F8, 9) (dual of [8388601, 8388544, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(857, 8388601, F8, 9) (dual of [8388601, 8388544, 10]-code), using
- net defined by OOA [i] based on linear OOA(857, 2097150, F8, 9, 9) (dual of [(2097150, 9), 18874293, 10]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(52, 52+9, large)-Net over F8 — Digital
Digital (52, 61, large)-net over F8, using
- 83 times duplication [i] based on digital (49, 58, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(858, large, F8, 9) (dual of [large, large−58, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(858, large, F8, 9) (dual of [large, large−58, 10]-code), using
(52, 52+9, large)-Net in Base 8 — Upper bound on s
There is no (52, 61, large)-net in base 8, because
- 7 times m-reduction [i] would yield (52, 54, large)-net in base 8, but