Best Known (211−25, 211, s)-Nets in Base 4
(211−25, 211, 349530)-Net over F4 — Constructive and digital
Digital (186, 211, 349530)-net over F4, using
- net defined by OOA [i] based on linear OOA(4211, 349530, F4, 25, 25) (dual of [(349530, 25), 8738039, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4211, 4194361, F4, 25) (dual of [4194361, 4194150, 26]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4209, 4194359, F4, 25) (dual of [4194359, 4194150, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 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([0,12]) ⊂ C([0,9]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4209, 4194359, F4, 25) (dual of [4194359, 4194150, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4211, 4194361, F4, 25) (dual of [4194361, 4194150, 26]-code), using
(211−25, 211, 1582330)-Net over F4 — Digital
Digital (186, 211, 1582330)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4211, 1582330, F4, 2, 25) (dual of [(1582330, 2), 3164449, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4211, 2097180, F4, 2, 25) (dual of [(2097180, 2), 4194149, 26]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4210, 2097180, F4, 2, 25) (dual of [(2097180, 2), 4194150, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4210, 4194360, F4, 25) (dual of [4194360, 4194150, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4209, 4194359, F4, 25) (dual of [4194359, 4194150, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4199, 4194305, F4, 25) (dual of [4194305, 4194106, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4155, 4194305, F4, 19) (dual of [4194305, 4194150, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 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([0,12]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4209, 4194359, F4, 25) (dual of [4194359, 4194150, 26]-code), using
- OOA 2-folding [i] based on linear OA(4210, 4194360, F4, 25) (dual of [4194360, 4194150, 26]-code), using
- 41 times duplication [i] based on linear OOA(4210, 2097180, F4, 2, 25) (dual of [(2097180, 2), 4194150, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4211, 2097180, F4, 2, 25) (dual of [(2097180, 2), 4194149, 26]-NRT-code), using
(211−25, 211, large)-Net in Base 4 — Upper bound on s
There is no (186, 211, large)-net in base 4, because
- 23 times m-reduction [i] would yield (186, 188, large)-net in base 4, but