Best Known (165, 165+21, s)-Nets in Base 4
(165, 165+21, 838860)-Net over F4 — Constructive and digital
Digital (165, 186, 838860)-net over F4, using
- 45 times duplication [i] based on digital (160, 181, 838860)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
(165, 165+21, 3592511)-Net over F4 — Digital
Digital (165, 186, 3592511)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4186, 3592511, F4, 2, 21) (dual of [(3592511, 2), 7184836, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4186, 4194303, F4, 2, 21) (dual of [(4194303, 2), 8388420, 22]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4185, 4194303, F4, 2, 21) (dual of [(4194303, 2), 8388421, 22]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4181, 4194301, F4, 2, 21) (dual of [(4194301, 2), 8388421, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4181, 8388602, F4, 21) (dual of [8388602, 8388421, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- OOA 2-folding [i] based on linear OA(4181, 8388602, F4, 21) (dual of [8388602, 8388421, 22]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4181, 4194301, F4, 2, 21) (dual of [(4194301, 2), 8388421, 22]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4185, 4194303, F4, 2, 21) (dual of [(4194303, 2), 8388421, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4186, 4194303, F4, 2, 21) (dual of [(4194303, 2), 8388420, 22]-NRT-code), using
(165, 165+21, large)-Net in Base 4 — Upper bound on s
There is no (165, 186, large)-net in base 4, because
- 19 times m-reduction [i] would yield (165, 167, large)-net in base 4, but