Best Known (128−20, 128, s)-Nets in Base 4
(128−20, 128, 6557)-Net over F4 — Constructive and digital
Digital (108, 128, 6557)-net over F4, using
- net defined by OOA [i] based on linear OOA(4128, 6557, F4, 20, 20) (dual of [(6557, 20), 131012, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4128, 65570, F4, 20) (dual of [65570, 65442, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 65575, F4, 20) (dual of [65575, 65447, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 65575, F4, 20) (dual of [65575, 65447, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4128, 65570, F4, 20) (dual of [65570, 65442, 21]-code), using
(128−20, 128, 44539)-Net over F4 — Digital
Digital (108, 128, 44539)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4128, 44539, F4, 20) (dual of [44539, 44411, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 65575, F4, 20) (dual of [65575, 65447, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 65575, F4, 20) (dual of [65575, 65447, 21]-code), using
(128−20, 128, large)-Net in Base 4 — Upper bound on s
There is no (108, 128, large)-net in base 4, because
- 18 times m-reduction [i] would yield (108, 110, large)-net in base 4, but