Best Known (254−40, 254, s)-Nets in Base 4
(254−40, 254, 3279)-Net over F4 — Constructive and digital
Digital (214, 254, 3279)-net over F4, using
- 41 times duplication [i] based on digital (213, 253, 3279)-net over F4, using
- t-expansion [i] based on digital (212, 253, 3279)-net over F4, using
- net defined by OOA [i] based on linear OOA(4253, 3279, F4, 41, 41) (dual of [(3279, 41), 134186, 42]-NRT-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4253, 65581, F4, 41) (dual of [65581, 65328, 42]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4251, 65579, F4, 41) (dual of [65579, 65328, 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(410, 42, F4, 5) (dual of [42, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4251, 65579, F4, 41) (dual of [65579, 65328, 42]-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4253, 65581, F4, 41) (dual of [65581, 65328, 42]-code), using
- net defined by OOA [i] based on linear OOA(4253, 3279, F4, 41, 41) (dual of [(3279, 41), 134186, 42]-NRT-code), using
- t-expansion [i] based on digital (212, 253, 3279)-net over F4, using
(254−40, 254, 51036)-Net over F4 — Digital
Digital (214, 254, 51036)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4254, 51036, F4, 40) (dual of [51036, 50782, 41]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(4254, 65598, F4, 40) (dual of [65598, 65344, 41]-code), using
(254−40, 254, large)-Net in Base 4 — Upper bound on s
There is no (214, 254, large)-net in base 4, because
- 38 times m-reduction [i] would yield (214, 216, large)-net in base 4, but