Best Known (105−20, 105, s)-Nets in Base 4
(105−20, 105, 1638)-Net over F4 — Constructive and digital
Digital (85, 105, 1638)-net over F4, using
- net defined by OOA [i] based on linear OOA(4105, 1638, F4, 20, 20) (dual of [(1638, 20), 32655, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4105, 16380, F4, 20) (dual of [16380, 16275, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4105, 16380, F4, 20) (dual of [16380, 16275, 21]-code), using
(105−20, 105, 8191)-Net over F4 — Digital
Digital (85, 105, 8191)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4105, 8191, F4, 2, 20) (dual of [(8191, 2), 16277, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4105, 16382, F4, 20) (dual of [16382, 16277, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- OOA 2-folding [i] based on linear OA(4105, 16382, F4, 20) (dual of [16382, 16277, 21]-code), using
(105−20, 105, 3165802)-Net in Base 4 — Upper bound on s
There is no (85, 105, 3165803)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1645 504685 940356 566771 557947 856311 473265 792936 248968 776001 317341 > 4105 [i]