Best Known (51, 51+6, s)-Nets in Base 4
(51, 51+6, 2796957)-Net over F4 — Constructive and digital
Digital (51, 57, 2796957)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 8, 756)-net over F4, using
- net defined by OOA [i] based on linear OOA(48, 756, F4, 3, 3) (dual of [(756, 3), 2260, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(48, 756, F4, 2, 3) (dual of [(756, 2), 1504, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(48, 756, F4, 3, 3) (dual of [(756, 3), 2260, 4]-NRT-code), using
- digital (43, 49, 2796201)-net over F4, using
- net defined by OOA [i] based on linear OOA(449, 2796201, F4, 6, 6) (dual of [(2796201, 6), 16777157, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(449, large, F4, 6) (dual of [large, large−49, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(449, large, F4, 6) (dual of [large, large−49, 7]-code), using
- net defined by OOA [i] based on linear OOA(449, 2796201, F4, 6, 6) (dual of [(2796201, 6), 16777157, 7]-NRT-code), using
- digital (5, 8, 756)-net over F4, using
(51, 51+6, large)-Net over F4 — Digital
Digital (51, 57, large)-net over F4, using
- 48 times duplication [i] based on digital (43, 49, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(449, large, F4, 6) (dual of [large, large−49, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(449, large, F4, 6) (dual of [large, large−49, 7]-code), using
(51, 51+6, large)-Net in Base 4 — Upper bound on s
There is no (51, 57, large)-net in base 4, because
- 4 times m-reduction [i] would yield (51, 53, large)-net in base 4, but