Best Known (147, 172, s)-Nets in Base 4
(147, 172, 21848)-Net over F4 — Constructive and digital
Digital (147, 172, 21848)-net over F4, using
- 43 times duplication [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
(147, 172, 131095)-Net over F4 — Digital
Digital (147, 172, 131095)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4172, 131095, F4, 2, 25) (dual of [(131095, 2), 262018, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4172, 262190, F4, 25) (dual of [262190, 262018, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(49, 45, F4, 5) (dual of [45, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(4172, 262190, F4, 25) (dual of [262190, 262018, 26]-code), using
(147, 172, large)-Net in Base 4 — Upper bound on s
There is no (147, 172, large)-net in base 4, because
- 23 times m-reduction [i] would yield (147, 149, large)-net in base 4, but