Best Known (134−12, 134, s)-Nets in Base 4
(134−12, 134, 2796200)-Net over F4 — Constructive and digital
Digital (122, 134, 2796200)-net over F4, using
- net defined by OOA [i] based on linear OOA(4134, 2796200, F4, 14, 12) (dual of [(2796200, 14), 39146666, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(4134, 8388601, F4, 2, 12) (dual of [(8388601, 2), 16777068, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4134, 8388602, F4, 2, 12) (dual of [(8388602, 2), 16777070, 13]-NRT-code), using
- trace code [i] based on linear OOA(1667, 4194301, F16, 2, 12) (dual of [(4194301, 2), 8388535, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1667, 8388602, F16, 12) (dual of [8388602, 8388535, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- OOA 2-folding [i] based on linear OA(1667, 8388602, F16, 12) (dual of [8388602, 8388535, 13]-code), using
- trace code [i] based on linear OOA(1667, 4194301, F16, 2, 12) (dual of [(4194301, 2), 8388535, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4134, 8388602, F4, 2, 12) (dual of [(8388602, 2), 16777070, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(4134, 8388601, F4, 2, 12) (dual of [(8388601, 2), 16777068, 13]-NRT-code), using
(134−12, 134, large)-Net over F4 — Digital
Digital (122, 134, large)-net over F4, using
- t-expansion [i] based on digital (121, 134, large)-net over F4, using
- 1 times m-reduction [i] based on digital (121, 135, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 14 times code embedding in larger space [i] 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]
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 1 times m-reduction [i] based on digital (121, 135, large)-net over F4, using
(134−12, 134, large)-Net in Base 4 — Upper bound on s
There is no (122, 134, large)-net in base 4, because
- 10 times m-reduction [i] would yield (122, 124, large)-net in base 4, but