Best Known (175, 175+23, s)-Nets in Base 4
(175, 175+23, 381306)-Net over F4 — Constructive and digital
Digital (175, 198, 381306)-net over F4, using
- net defined by OOA [i] based on linear OOA(4198, 381306, F4, 23, 23) (dual of [(381306, 23), 8769840, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4198, 4194367, F4, 23) (dual of [4194367, 4194169, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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]
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4198, 4194367, F4, 23) (dual of [4194367, 4194169, 24]-code), using
(175, 175+23, 2097184)-Net over F4 — Digital
Digital (175, 198, 2097184)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4198, 2097184, F4, 2, 23) (dual of [(2097184, 2), 4194170, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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]
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4198, 4194368, F4, 23) (dual of [4194368, 4194170, 24]-code), using
(175, 175+23, large)-Net in Base 4 — Upper bound on s
There is no (175, 198, large)-net in base 4, because
- 21 times m-reduction [i] would yield (175, 177, large)-net in base 4, but