Best Known (250−41, 250, s)-Nets in Base 4
(250−41, 250, 3278)-Net over F4 — Constructive and digital
Digital (209, 250, 3278)-net over F4, using
- 44 times duplication [i] based on digital (205, 246, 3278)-net over F4, using
- net defined by OOA [i] based on linear OOA(4246, 3278, F4, 41, 41) (dual of [(3278, 41), 134152, 42]-NRT-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4246, 65561, F4, 41) (dual of [65561, 65315, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4246, 65565, F4, 41) (dual of [65565, 65319, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(4241, 65536, F4, 41) (dual of [65536, 65295, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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(40) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(4246, 65565, F4, 41) (dual of [65565, 65319, 42]-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4246, 65561, F4, 41) (dual of [65561, 65315, 42]-code), using
- net defined by OOA [i] based on linear OOA(4246, 3278, F4, 41, 41) (dual of [(3278, 41), 134152, 42]-NRT-code), using
(250−41, 250, 35797)-Net over F4 — Digital
Digital (209, 250, 35797)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4250, 35797, F4, 41) (dual of [35797, 35547, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4250, 65573, F4, 41) (dual of [65573, 65323, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(49, 36, F4, 5) (dual of [36, 27, 6]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4250, 65573, F4, 41) (dual of [65573, 65323, 42]-code), using
(250−41, 250, large)-Net in Base 4 — Upper bound on s
There is no (209, 250, large)-net in base 4, because
- 39 times m-reduction [i] would yield (209, 211, large)-net in base 4, but