Best Known (220, 249, s)-Nets in Base 4
(220, 249, 299607)-Net over F4 — Constructive and digital
Digital (220, 249, 299607)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (203, 232, 299593)-net over F4, using
- net defined by OOA [i] based on linear OOA(4232, 299593, F4, 29, 29) (dual of [(299593, 29), 8687965, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4232, 4194303, F4, 29) (dual of [4194303, 4194071, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4232, 4194303, F4, 29) (dual of [4194303, 4194071, 30]-code), using
- net defined by OOA [i] based on linear OOA(4232, 299593, F4, 29, 29) (dual of [(299593, 29), 8687965, 30]-NRT-code), using
- digital (3, 17, 14)-net over F4, using
(220, 249, 1843898)-Net over F4 — Digital
Digital (220, 249, 1843898)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4249, 1843898, F4, 2, 29) (dual of [(1843898, 2), 3687547, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4249, 2097188, F4, 2, 29) (dual of [(2097188, 2), 4194127, 30]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4245, 2097186, F4, 2, 29) (dual of [(2097186, 2), 4194127, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4245, 4194372, F4, 29) (dual of [4194372, 4194127, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(413, 68, F4, 6) (dual of [68, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 70, F4, 6) (dual of [70, 57, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4245, 4194372, F4, 29) (dual of [4194372, 4194127, 30]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4245, 2097186, F4, 2, 29) (dual of [(2097186, 2), 4194127, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4249, 2097188, F4, 2, 29) (dual of [(2097188, 2), 4194127, 30]-NRT-code), using
(220, 249, large)-Net in Base 4 — Upper bound on s
There is no (220, 249, large)-net in base 4, because
- 27 times m-reduction [i] would yield (220, 222, large)-net in base 4, but