Best Known (66, 73, s)-Nets in Base 4
(66, 73, 2830766)-Net over F4 — Constructive and digital
Digital (66, 73, 2830766)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 2830766, F4, 9, 7) (dual of [(2830766, 9), 25476821, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(473, 2830767, F4, 3, 7) (dual of [(2830767, 3), 8492228, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(412, 34566, F4, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(412, 34566, F4, 3) (dual of [34566, 34554, 4]-code or 34566-cap in PG(11,4)), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
- linear OOA(461, 2796201, F4, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- linear OOA(412, 34566, F4, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(473, 2830767, F4, 3, 7) (dual of [(2830767, 3), 8492228, 8]-NRT-code), using
(66, 73, large)-Net over F4 — Digital
Digital (66, 73, large)-net over F4, using
- 41 times duplication [i] based on digital (65, 72, large)-net over F4, using
- t-expansion [i] based on digital (64, 72, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(472, large, F4, 8) (dual of [large, large−72, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(472, large, F4, 8) (dual of [large, large−72, 9]-code), using
- t-expansion [i] based on digital (64, 72, large)-net over F4, using
(66, 73, large)-Net in Base 4 — Upper bound on s
There is no (66, 73, large)-net in base 4, because
- 5 times m-reduction [i] would yield (66, 68, large)-net in base 4, but