Best Known (248−17, 248, s)-Nets in Base 4
(248−17, 248, 4210689)-Net over F4 — Constructive and digital
Digital (231, 248, 4210689)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (44, 52, 16389)-net over F4, using
- net defined by OOA [i] based on linear OOA(452, 16389, F4, 8, 8) (dual of [(16389, 8), 131060, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(452, 65556, F4, 8) (dual of [65556, 65504, 9]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(433, 65537, F4, 5) (dual of [65537, 65504, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,2], and minimum distance d ≥ |{−4,−2,0,2,4}|+1 = 6 (BCH-bound) [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(452, 65556, F4, 8) (dual of [65556, 65504, 9]-code), using
- net defined by OOA [i] based on linear OOA(452, 16389, F4, 8, 8) (dual of [(16389, 8), 131060, 9]-NRT-code), using
- digital (179, 196, 4194300)-net over F4, using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- digital (44, 52, 16389)-net over F4, using
(248−17, 248, large)-Net over F4 — Digital
Digital (231, 248, large)-net over F4, using
- 9 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 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]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
(248−17, 248, large)-Net in Base 4 — Upper bound on s
There is no (231, 248, large)-net in base 4, because
- 15 times m-reduction [i] would yield (231, 233, large)-net in base 4, but