Best Known (111, 111+14, s)-Nets in Base 4
(111, 111+14, 1198371)-Net over F4 — Constructive and digital
Digital (111, 125, 1198371)-net over F4, using
- 44 times duplication [i] based on digital (107, 121, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 1198371, F4, 14, 14) (dual of [(1198371, 14), 16777073, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4121, 8388597, F4, 14) (dual of [8388597, 8388476, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4121, 8388597, F4, 14) (dual of [8388597, 8388476, 15]-code), using
- net defined by OOA [i] based on linear OOA(4121, 1198371, F4, 14, 14) (dual of [(1198371, 14), 16777073, 15]-NRT-code), using
(111, 111+14, 4194303)-Net over F4 — Digital
Digital (111, 125, 4194303)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4125, 4194303, F4, 2, 14) (dual of [(4194303, 2), 8388481, 15]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4121, 4194301, F4, 2, 14) (dual of [(4194301, 2), 8388481, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4121, 8388602, F4, 14) (dual of [8388602, 8388481, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- OOA 2-folding [i] based on linear OA(4121, 8388602, F4, 14) (dual of [8388602, 8388481, 15]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4121, 4194301, F4, 2, 14) (dual of [(4194301, 2), 8388481, 15]-NRT-code), using
(111, 111+14, large)-Net in Base 4 — Upper bound on s
There is no (111, 125, large)-net in base 4, because
- 12 times m-reduction [i] would yield (111, 113, large)-net in base 4, but