Best Known (66−6, 66, s)-Nets in Base 5
(66−6, 66, 8033808)-Net over F5 — Constructive and digital
Digital (60, 66, 8033808)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (8, 10, 2441406)-net over F5, using
- digital (12, 15, 2796201)-net over F5, using
- s-reduction based on digital (12, 15, 3050326)-net over F5, using
- net defined by OOA [i] based on linear OOA(515, 3050326, F5, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(515, 3050326, F5, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(515, 3050326, F5, 3) (dual of [3050326, 3050311, 4]-code or 3050326-cap in PG(14,5)), using
- appending kth column [i] based on linear OOA(515, 3050326, F5, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(515, 3050326, F5, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
- s-reduction based on digital (12, 15, 3050326)-net over F5, using
- digital (35, 41, 2796201)-net over F5, using
- net defined by OOA [i] based on linear OOA(541, 2796201, F5, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(541, large, F5, 6) (dual of [large, large−41, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(541, large, F5, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(541, 2796201, F5, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
(66−6, 66, large)-Net over F5 — Digital
Digital (60, 66, large)-net over F5, using
- 54 times duplication [i] based on digital (56, 62, large)-net over F5, using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
(66−6, 66, large)-Net in Base 5 — Upper bound on s
There is no (60, 66, large)-net in base 5, because
- 4 times m-reduction [i] would yield (60, 62, large)-net in base 5, but