Best Known (147−19, 147, s)-Nets in Base 4
(147−19, 147, 116512)-Net over F4 — Constructive and digital
Digital (128, 147, 116512)-net over F4, using
- 41 times duplication [i] based on digital (127, 146, 116512)-net over F4, using
- net defined by OOA [i] based on linear OOA(4146, 116512, F4, 19, 19) (dual of [(116512, 19), 2213582, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4146, 1048609, F4, 19) (dual of [1048609, 1048463, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4146, 1048611, F4, 19) (dual of [1048611, 1048465, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4146, 1048611, F4, 19) (dual of [1048611, 1048465, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4146, 1048609, F4, 19) (dual of [1048609, 1048463, 20]-code), using
- net defined by OOA [i] based on linear OOA(4146, 116512, F4, 19, 19) (dual of [(116512, 19), 2213582, 20]-NRT-code), using
(147−19, 147, 524306)-Net over F4 — Digital
Digital (128, 147, 524306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4147, 524306, F4, 2, 19) (dual of [(524306, 2), 1048465, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4147, 1048612, F4, 19) (dual of [1048612, 1048465, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4146, 1048611, F4, 19) (dual of [1048611, 1048465, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4146, 1048611, F4, 19) (dual of [1048611, 1048465, 20]-code), using
- OOA 2-folding [i] based on linear OA(4147, 1048612, F4, 19) (dual of [1048612, 1048465, 20]-code), using
(147−19, 147, large)-Net in Base 4 — Upper bound on s
There is no (128, 147, large)-net in base 4, because
- 17 times m-reduction [i] would yield (128, 130, large)-net in base 4, but