Best Known (112, 112+22, s)-Nets in Base 4
(112, 112+22, 5960)-Net over F4 — Constructive and digital
Digital (112, 134, 5960)-net over F4, using
- net defined by OOA [i] based on linear OOA(4134, 5960, F4, 22, 22) (dual of [(5960, 22), 130986, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4134, 65560, F4, 22) (dual of [65560, 65426, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4134, 65565, F4, 22) (dual of [65565, 65431, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4134, 65565, F4, 22) (dual of [65565, 65431, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4134, 65560, F4, 22) (dual of [65560, 65426, 23]-code), using
(112, 112+22, 32782)-Net over F4 — Digital
Digital (112, 134, 32782)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4134, 32782, F4, 2, 22) (dual of [(32782, 2), 65430, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4134, 65564, F4, 22) (dual of [65564, 65430, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4134, 65565, F4, 22) (dual of [65565, 65431, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4134, 65565, F4, 22) (dual of [65565, 65431, 23]-code), using
- OOA 2-folding [i] based on linear OA(4134, 65564, F4, 22) (dual of [65564, 65430, 23]-code), using
(112, 112+22, large)-Net in Base 4 — Upper bound on s
There is no (112, 134, large)-net in base 4, because
- 20 times m-reduction [i] would yield (112, 114, large)-net in base 4, but