Best Known (126−21, 126, s)-Nets in Base 4
(126−21, 126, 6556)-Net over F4 — Constructive and digital
Digital (105, 126, 6556)-net over F4, using
- net defined by OOA [i] based on linear OOA(4126, 6556, F4, 21, 21) (dual of [(6556, 21), 137550, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4126, 65561, F4, 21) (dual of [65561, 65435, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [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(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4126, 65561, F4, 21) (dual of [65561, 65435, 22]-code), using
(126−21, 126, 32782)-Net over F4 — Digital
Digital (105, 126, 32782)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4126, 32782, F4, 2, 21) (dual of [(32782, 2), 65438, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4126, 65564, F4, 21) (dual of [65564, 65438, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [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(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- OOA 2-folding [i] based on linear OA(4126, 65564, F4, 21) (dual of [65564, 65438, 22]-code), using
(126−21, 126, large)-Net in Base 4 — Upper bound on s
There is no (105, 126, large)-net in base 4, because
- 19 times m-reduction [i] would yield (105, 107, large)-net in base 4, but