Best Known (249−27, 249, s)-Nets in Base 4
(249−27, 249, 645277)-Net over F4 — Constructive and digital
Digital (222, 249, 645277)-net over F4, using
- 48 times duplication [i] based on digital (214, 241, 645277)-net over F4, using
- net defined by OOA [i] based on linear OOA(4241, 645277, F4, 27, 27) (dual of [(645277, 27), 17422238, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4241, 8388602, F4, 27) (dual of [8388602, 8388361, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4241, large, F4, 27) (dual of [large, large−241, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4241, large, F4, 27) (dual of [large, large−241, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4241, 8388602, F4, 27) (dual of [8388602, 8388361, 28]-code), using
- net defined by OOA [i] based on linear OOA(4241, 645277, F4, 27, 27) (dual of [(645277, 27), 17422238, 28]-NRT-code), using
(249−27, 249, 4194305)-Net over F4 — Digital
Digital (222, 249, 4194305)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4249, 4194305, F4, 2, 27) (dual of [(4194305, 2), 8388361, 28]-NRT-code), using
- 4 times NRT-code embedding in larger space [i] based on linear OOA(4241, 4194301, F4, 2, 27) (dual of [(4194301, 2), 8388361, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4241, 8388602, F4, 27) (dual of [8388602, 8388361, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4241, large, F4, 27) (dual of [large, large−241, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4241, large, F4, 27) (dual of [large, large−241, 28]-code), using
- OOA 2-folding [i] based on linear OA(4241, 8388602, F4, 27) (dual of [8388602, 8388361, 28]-code), using
- 4 times NRT-code embedding in larger space [i] based on linear OOA(4241, 4194301, F4, 2, 27) (dual of [(4194301, 2), 8388361, 28]-NRT-code), using
(249−27, 249, large)-Net in Base 4 — Upper bound on s
There is no (222, 249, large)-net in base 4, because
- 25 times m-reduction [i] would yield (222, 224, large)-net in base 4, but