Best Known (169−24, 169, s)-Nets in Base 4
(169−24, 169, 21848)-Net over F4 — Constructive and digital
Digital (145, 169, 21848)-net over F4, using
- t-expansion [i] based on digital (144, 169, 21848)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 21848, F4, 25, 25) (dual of [(21848, 25), 546031, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4169, 262177, F4, 25) (dual of [262177, 262008, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4168, 262176, F4, 25) (dual of [262176, 262008, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4168, 262176, F4, 25) (dual of [262176, 262008, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4169, 262177, F4, 25) (dual of [262177, 262008, 26]-code), using
- net defined by OOA [i] based on linear OOA(4169, 21848, F4, 25, 25) (dual of [(21848, 25), 546031, 26]-NRT-code), using
(169−24, 169, 131088)-Net over F4 — Digital
Digital (145, 169, 131088)-net over F4, using
- 41 times duplication [i] based on digital (144, 168, 131088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4168, 131088, F4, 2, 24) (dual of [(131088, 2), 262008, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4168, 262176, F4, 24) (dual of [262176, 262008, 25]-code), using
- strength reduction [i] based on linear OA(4168, 262176, F4, 25) (dual of [262176, 262008, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- strength reduction [i] based on linear OA(4168, 262176, F4, 25) (dual of [262176, 262008, 26]-code), using
- OOA 2-folding [i] based on linear OA(4168, 262176, F4, 24) (dual of [262176, 262008, 25]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4168, 131088, F4, 2, 24) (dual of [(131088, 2), 262008, 25]-NRT-code), using
(169−24, 169, large)-Net in Base 4 — Upper bound on s
There is no (145, 169, large)-net in base 4, because
- 22 times m-reduction [i] would yield (145, 147, large)-net in base 4, but