Best Known (89, 89+9, s)-Nets in Base 4
(89, 89+9, 4194300)-Net over F4 — Constructive and digital
Digital (89, 98, 4194300)-net over F4, using
- net defined by OOA [i] based on linear OOA(498, 4194300, F4, 10, 9) (dual of [(4194300, 10), 41942902, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(498, 8388601, F4, 2, 9) (dual of [(8388601, 2), 16777104, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(498, 8388602, F4, 2, 9) (dual of [(8388602, 2), 16777106, 10]-NRT-code), using
- trace code [i] based on linear OOA(1649, 4194301, F16, 2, 9) (dual of [(4194301, 2), 8388553, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1649, 8388602, F16, 9) (dual of [8388602, 8388553, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, 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(1649, large, F16, 9) (dual of [large, large−49, 10]-code), using
- OOA 2-folding [i] based on linear OA(1649, 8388602, F16, 9) (dual of [8388602, 8388553, 10]-code), using
- trace code [i] based on linear OOA(1649, 4194301, F16, 2, 9) (dual of [(4194301, 2), 8388553, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(498, 8388602, F4, 2, 9) (dual of [(8388602, 2), 16777106, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(498, 8388601, F4, 2, 9) (dual of [(8388601, 2), 16777104, 10]-NRT-code), using
(89, 89+9, large)-Net over F4 — Digital
Digital (89, 98, large)-net over F4, using
- 46 times duplication [i] based on digital (83, 92, large)-net over F4, using
- t-expansion [i] based on digital (82, 92, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(492, large, F4, 10) (dual of [large, large−92, 11]-code), using
- 7 times code embedding in larger space [i] based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 7 times code embedding in larger space [i] based on linear OA(485, large, F4, 10) (dual of [large, large−85, 11]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(492, large, F4, 10) (dual of [large, large−92, 11]-code), using
- t-expansion [i] based on digital (82, 92, large)-net over F4, using
(89, 89+9, large)-Net in Base 4 — Upper bound on s
There is no (89, 98, large)-net in base 4, because
- 7 times m-reduction [i] would yield (89, 91, large)-net in base 4, but