Best Known (220, 220+28, s)-Nets in Base 4
(220, 220+28, 299607)-Net over F4 — Constructive and digital
Digital (220, 248, 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, 231, 299593)-net over F4, using
- net defined by OOA [i] based on linear OOA(4231, 299593, F4, 28, 28) (dual of [(299593, 28), 8388373, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4231, 4194302, F4, 28) (dual of [4194302, 4194071, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 4194303, F4, 28) (dual of [4194303, 4194072, 29]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 4194303, F4, 28) (dual of [4194303, 4194072, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4231, 4194302, F4, 28) (dual of [4194302, 4194071, 29]-code), using
- net defined by OOA [i] based on linear OOA(4231, 299593, F4, 28, 28) (dual of [(299593, 28), 8388373, 29]-NRT-code), using
- digital (3, 17, 14)-net over F4, using
(220, 220+28, 2097193)-Net over F4 — Digital
Digital (220, 248, 2097193)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4248, 2097193, F4, 2, 28) (dual of [(2097193, 2), 4194138, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4248, 4194386, F4, 28) (dual of [4194386, 4194138, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [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(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(416, 82, F4, 6) (dual of [82, 66, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 85, F4, 6) (dual of [85, 69, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(4248, 4194386, F4, 28) (dual of [4194386, 4194138, 29]-code), using
(220, 220+28, large)-Net in Base 4 — Upper bound on s
There is no (220, 248, large)-net in base 4, because
- 26 times m-reduction [i] would yield (220, 222, large)-net in base 4, but