Best Known (138−15, 138, s)-Nets in Base 4
(138−15, 138, 1198371)-Net over F4 — Constructive and digital
Digital (123, 138, 1198371)-net over F4, using
- 45 times duplication [i] based on digital (118, 133, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
(138−15, 138, 4194303)-Net over F4 — Digital
Digital (123, 138, 4194303)-net over F4, using
- 41 times duplication [i] based on digital (122, 137, 4194303)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4137, 4194303, F4, 2, 15) (dual of [(4194303, 2), 8388469, 16]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4133, 4194301, F4, 2, 15) (dual of [(4194301, 2), 8388469, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4133, 8388602, F4, 15) (dual of [8388602, 8388469, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- OOA 2-folding [i] based on linear OA(4133, 8388602, F4, 15) (dual of [8388602, 8388469, 16]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4133, 4194301, F4, 2, 15) (dual of [(4194301, 2), 8388469, 16]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4137, 4194303, F4, 2, 15) (dual of [(4194303, 2), 8388469, 16]-NRT-code), using
(138−15, 138, large)-Net in Base 4 — Upper bound on s
There is no (123, 138, large)-net in base 4, because
- 13 times m-reduction [i] would yield (123, 125, large)-net in base 4, but