Best Known (52, 52+7, s)-Nets in Base 8
(52, 52+7, 5637575)-Net over F8 — Constructive and digital
Digital (52, 59, 5637575)-net over F8, using
- net defined by OOA [i] based on linear OOA(859, 5637575, F8, 9, 7) (dual of [(5637575, 9), 50738116, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(859, 5637576, F8, 3, 7) (dual of [(5637576, 3), 16912669, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(89, 45174, F8, 3, 3) (dual of [(45174, 3), 135513, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(89, 45174, F8, 2, 3) (dual of [(45174, 2), 90339, 4]-NRT-code), using
- linear OOA(850, 5592402, F8, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- trace code [i] based on linear OOA(6425, 2796201, F64, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- linear OOA(89, 45174, F8, 3, 3) (dual of [(45174, 3), 135513, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(859, 5637576, F8, 3, 7) (dual of [(5637576, 3), 16912669, 8]-NRT-code), using
(52, 52+7, large)-Net over F8 — Digital
Digital (52, 59, large)-net over F8, using
- 81 times duplication [i] based on digital (51, 58, large)-net over F8, using
- t-expansion [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
- t-expansion [i] based on digital (49, 58, large)-net over F8, using
(52, 52+7, large)-Net in Base 8 — Upper bound on s
There is no (52, 59, large)-net in base 8, because
- 5 times m-reduction [i] would yield (52, 54, large)-net in base 8, but