Best Known (111−20, 111, s)-Nets in Base 4
(111−20, 111, 1641)-Net over F4 — Constructive and digital
Digital (91, 111, 1641)-net over F4, using
- net defined by OOA [i] based on linear OOA(4111, 1641, F4, 20, 20) (dual of [(1641, 20), 32709, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4111, 16410, F4, 20) (dual of [16410, 16299, 21]-code), using
- strength reduction [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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
- strength reduction [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- OA 10-folding and stacking [i] based on linear OA(4111, 16410, F4, 20) (dual of [16410, 16299, 21]-code), using
(111−20, 111, 12016)-Net over F4 — Digital
Digital (91, 111, 12016)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 12016, F4, 20) (dual of [12016, 11905, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 16410, F4, 20) (dual of [16410, 16299, 21]-code), using
- strength reduction [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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
- strength reduction [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 16410, F4, 20) (dual of [16410, 16299, 21]-code), using
(111−20, 111, 7273115)-Net in Base 4 — Upper bound on s
There is no (91, 111, 7273116)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 739991 274640 928049 562757 635674 185037 714125 789964 145843 257211 819645 > 4111 [i]