Best Known (186, 186+24, s)-Nets in Base 4
(186, 186+24, 349530)-Net over F4 — Constructive and digital
Digital (186, 210, 349530)-net over F4, using
- 1 times m-reduction [i] based on 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
- net defined by OOA [i] based on linear OOA(4211, 349530, F4, 25, 25) (dual of [(349530, 25), 8738039, 26]-NRT-code), using
(186, 186+24, 2097185)-Net over F4 — Digital
Digital (186, 210, 2097185)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4210, 2097185, F4, 2, 24) (dual of [(2097185, 2), 4194160, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4210, 4194370, F4, 24) (dual of [4194370, 4194160, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(411, 66, F4, 5) (dual of [66, 55, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(47, 64, F4, 3) (dual of [64, 57, 4]-code or 64-cap in PG(6,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4210, 4194370, F4, 24) (dual of [4194370, 4194160, 25]-code), using
(186, 186+24, large)-Net in Base 4 — Upper bound on s
There is no (186, 210, large)-net in base 4, because
- 22 times m-reduction [i] would yield (186, 188, large)-net in base 4, but