Best Known (145, 145+22, s)-Nets in Base 4
(145, 145+22, 95328)-Net over F4 — Constructive and digital
Digital (145, 167, 95328)-net over F4, using
- 41 times duplication [i] based on digital (144, 166, 95328)-net over F4, using
- net defined by OOA [i] based on linear OOA(4166, 95328, F4, 22, 22) (dual of [(95328, 22), 2097050, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4166, 1048608, F4, 22) (dual of [1048608, 1048442, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4166, 1048611, F4, 22) (dual of [1048611, 1048445, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4166, 1048611, F4, 22) (dual of [1048611, 1048445, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4166, 1048608, F4, 22) (dual of [1048608, 1048442, 23]-code), using
- net defined by OOA [i] based on linear OOA(4166, 95328, F4, 22, 22) (dual of [(95328, 22), 2097050, 23]-NRT-code), using
(145, 145+22, 447194)-Net over F4 — Digital
Digital (145, 167, 447194)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4167, 447194, F4, 2, 22) (dual of [(447194, 2), 894221, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 524306, F4, 2, 22) (dual of [(524306, 2), 1048445, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4167, 1048612, F4, 22) (dual of [1048612, 1048445, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4166, 1048611, F4, 22) (dual of [1048611, 1048445, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-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(21) ⊂ Ce(17) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4166, 1048611, F4, 22) (dual of [1048611, 1048445, 23]-code), using
- OOA 2-folding [i] based on linear OA(4167, 1048612, F4, 22) (dual of [1048612, 1048445, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 524306, F4, 2, 22) (dual of [(524306, 2), 1048445, 23]-NRT-code), using
(145, 145+22, large)-Net in Base 4 — Upper bound on s
There is no (145, 167, large)-net in base 4, because
- 20 times m-reduction [i] would yield (145, 147, large)-net in base 4, but