Best Known (136, 136+21, s)-Nets in Base 4
(136, 136+21, 104861)-Net over F4 — Constructive and digital
Digital (136, 157, 104861)-net over F4, using
- 41 times duplication [i] based on digital (135, 156, 104861)-net over F4, using
- net defined by OOA [i] based on linear OOA(4156, 104861, F4, 21, 21) (dual of [(104861, 21), 2201925, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4156, 1048611, F4, 21) (dual of [1048611, 1048455, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(20) ⊂ Ce(16) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(4156, 1048611, F4, 21) (dual of [1048611, 1048455, 22]-code), using
- net defined by OOA [i] based on linear OOA(4156, 104861, F4, 21, 21) (dual of [(104861, 21), 2201925, 22]-NRT-code), using
(136, 136+21, 384945)-Net over F4 — Digital
Digital (136, 157, 384945)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4157, 384945, F4, 2, 21) (dual of [(384945, 2), 769733, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 524306, F4, 2, 21) (dual of [(524306, 2), 1048455, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4157, 1048612, F4, 21) (dual of [1048612, 1048455, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4156, 1048611, F4, 21) (dual of [1048611, 1048455, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(20) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4156, 1048611, F4, 21) (dual of [1048611, 1048455, 22]-code), using
- OOA 2-folding [i] based on linear OA(4157, 1048612, F4, 21) (dual of [1048612, 1048455, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 524306, F4, 2, 21) (dual of [(524306, 2), 1048455, 22]-NRT-code), using
(136, 136+21, large)-Net in Base 4 — Upper bound on s
There is no (136, 157, large)-net in base 4, because
- 19 times m-reduction [i] would yield (136, 138, large)-net in base 4, but