Best Known (65, 65+8, s)-Nets in Base 4
(65, 65+8, 2097150)-Net over F4 — Constructive and digital
Digital (65, 73, 2097150)-net over F4, using
- t-expansion [i] based on digital (64, 73, 2097150)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 2097150, F4, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(473, 8388601, F4, 9) (dual of [8388601, 8388528, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(473, 8388601, F4, 9) (dual of [8388601, 8388528, 10]-code), using
- net defined by OOA [i] based on linear OOA(473, 2097150, F4, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
(65, 65+8, large)-Net over F4 — Digital
Digital (65, 73, large)-net over F4, using
- 41 times duplication [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
(65, 65+8, large)-Net in Base 4 — Upper bound on s
There is no (65, 73, large)-net in base 4, because
- 6 times m-reduction [i] would yield (65, 67, large)-net in base 4, but