Best Known (219, 219+40, s)-Nets in Base 4
(219, 219+40, 3280)-Net over F4 — Constructive and digital
Digital (219, 259, 3280)-net over F4, using
- 42 times duplication [i] based on digital (217, 257, 3280)-net over F4, using
- t-expansion [i] based on digital (216, 257, 3280)-net over F4, using
- net defined by OOA [i] based on linear OOA(4257, 3280, F4, 41, 41) (dual of [(3280, 41), 134223, 42]-NRT-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4257, 65601, F4, 41) (dual of [65601, 65344, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [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(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- OOA 20-folding and stacking with additional row [i] based on linear OA(4257, 65601, F4, 41) (dual of [65601, 65344, 42]-code), using
- net defined by OOA [i] based on linear OOA(4257, 3280, F4, 41, 41) (dual of [(3280, 41), 134223, 42]-NRT-code), using
- t-expansion [i] based on digital (216, 257, 3280)-net over F4, using
(219, 219+40, 61254)-Net over F4 — Digital
Digital (219, 259, 61254)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4259, 61254, F4, 40) (dual of [61254, 60995, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 65603, F4, 40) (dual of [65603, 65344, 41]-code), using
- 5 times code embedding in larger space [i] based on linear OA(4254, 65598, F4, 40) (dual of [65598, 65344, 41]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [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(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(4254, 65598, F4, 40) (dual of [65598, 65344, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4259, 65603, F4, 40) (dual of [65603, 65344, 41]-code), using
(219, 219+40, large)-Net in Base 4 — Upper bound on s
There is no (219, 259, large)-net in base 4, because
- 38 times m-reduction [i] would yield (219, 221, large)-net in base 4, but