Best Known (253−26, 253, s)-Nets in Base 4
(253−26, 253, 645304)-Net over F4 — Constructive and digital
Digital (227, 253, 645304)-net over F4, using
- 41 times duplication [i] based on digital (226, 252, 645304)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 23, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- digital (203, 229, 645277)-net over F4, using
- net defined by OOA [i] based on linear OOA(4229, 645277, F4, 26, 26) (dual of [(645277, 26), 16776973, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4229, 8388601, F4, 26) (dual of [8388601, 8388372, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4229, 8388601, F4, 26) (dual of [8388601, 8388372, 27]-code), using
- net defined by OOA [i] based on linear OOA(4229, 645277, F4, 26, 26) (dual of [(645277, 26), 16776973, 27]-NRT-code), using
- digital (10, 23, 27)-net over F4, using
- (u, u+v)-construction [i] based on
(253−26, 253, 6852834)-Net over F4 — Digital
Digital (227, 253, 6852834)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4253, 6852834, F4, 26) (dual of [6852834, 6852581, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, large, F4, 26) (dual of [large, large−253, 27]-code), using
- strength reduction [i] based on linear OA(4253, large, F4, 29) (dual of [large, large−253, 30]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- strength reduction [i] based on linear OA(4253, large, F4, 29) (dual of [large, large−253, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, large, F4, 26) (dual of [large, large−253, 27]-code), using
(253−26, 253, large)-Net in Base 4 — Upper bound on s
There is no (227, 253, large)-net in base 4, because
- 24 times m-reduction [i] would yield (227, 229, large)-net in base 4, but