Best Known (249−26, 249, s)-Nets in Base 4
(249−26, 249, 645298)-Net over F4 — Constructive and digital
Digital (223, 249, 645298)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-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 (7, 20, 21)-net over F4, using
(249−26, 249, 5439094)-Net over F4 — Digital
Digital (223, 249, 5439094)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4249, 5439094, F4, 26) (dual of [5439094, 5438845, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4249, large, F4, 26) (dual of [large, large−249, 27]-code), using
- 20 times code embedding in larger space [i] 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]
- 20 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4249, large, F4, 26) (dual of [large, large−249, 27]-code), using
(249−26, 249, large)-Net in Base 4 — Upper bound on s
There is no (223, 249, large)-net in base 4, because
- 24 times m-reduction [i] would yield (223, 225, large)-net in base 4, but