Best Known (155, 155+29, s)-Nets in Base 4
(155, 155+29, 4690)-Net over F4 — Constructive and digital
Digital (155, 184, 4690)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (140, 169, 4681)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 4681, F4, 29, 29) (dual of [(4681, 29), 135580, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4169, 65535, F4, 29) (dual of [65535, 65366, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4169, 65535, F4, 29) (dual of [65535, 65366, 30]-code), using
- net defined by OOA [i] based on linear OOA(4169, 4681, F4, 29, 29) (dual of [(4681, 29), 135580, 30]-NRT-code), using
- digital (1, 15, 9)-net over F4, using
(155, 155+29, 43823)-Net over F4 — Digital
Digital (155, 184, 43823)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4184, 43823, F4, 29) (dual of [43823, 43639, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4184, 65591, F4, 29) (dual of [65591, 65407, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4182, 65589, F4, 29) (dual of [65589, 65407, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4182, 65589, F4, 29) (dual of [65589, 65407, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4184, 65591, F4, 29) (dual of [65591, 65407, 30]-code), using
(155, 155+29, large)-Net in Base 4 — Upper bound on s
There is no (155, 184, large)-net in base 4, because
- 27 times m-reduction [i] would yield (155, 157, large)-net in base 4, but