Best Known (149, 149+21, s)-Nets in Base 4
(149, 149+21, 419432)-Net over F4 — Constructive and digital
Digital (149, 170, 419432)-net over F4, using
- 41 times duplication [i] based on digital (148, 169, 419432)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 419432, F4, 21, 21) (dual of [(419432, 21), 8807903, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4169, 4194321, F4, 21) (dual of [4194321, 4194152, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4169, 4194325, F4, 21) (dual of [4194325, 4194156, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4169, 4194325, F4, 21) (dual of [4194325, 4194156, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4169, 4194321, F4, 21) (dual of [4194321, 4194152, 22]-code), using
- net defined by OOA [i] based on linear OOA(4169, 419432, F4, 21, 21) (dual of [(419432, 21), 8807903, 22]-NRT-code), using
(149, 149+21, 1398110)-Net over F4 — Digital
Digital (149, 170, 1398110)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4170, 1398110, F4, 3, 21) (dual of [(1398110, 3), 4194160, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4170, 4194330, F4, 21) (dual of [4194330, 4194160, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(44, 26, F4, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- OOA 3-folding [i] based on linear OA(4170, 4194330, F4, 21) (dual of [4194330, 4194160, 22]-code), using
(149, 149+21, large)-Net in Base 4 — Upper bound on s
There is no (149, 170, large)-net in base 4, because
- 19 times m-reduction [i] would yield (149, 151, large)-net in base 4, but