Best Known (164, 164+22, s)-Nets in Base 4
(164, 164+22, 381305)-Net over F4 — Constructive and digital
Digital (164, 186, 381305)-net over F4, using
- 41 times duplication [i] based on digital (163, 185, 381305)-net over F4, using
- net defined by OOA [i] based on linear OOA(4185, 381305, F4, 22, 22) (dual of [(381305, 22), 8388525, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4185, 4194355, F4, 22) (dual of [4194355, 4194170, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4185, 4194355, F4, 22) (dual of [4194355, 4194170, 23]-code), using
- net defined by OOA [i] based on linear OOA(4185, 381305, F4, 22, 22) (dual of [(381305, 22), 8388525, 23]-NRT-code), using
(164, 164+22, 1788819)-Net over F4 — Digital
Digital (164, 186, 1788819)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4186, 1788819, F4, 2, 22) (dual of [(1788819, 2), 3577452, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4186, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194170, 23]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4185, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194171, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- 41 times duplication [i] based on linear OOA(4185, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194171, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4186, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194170, 23]-NRT-code), using
(164, 164+22, large)-Net in Base 4 — Upper bound on s
There is no (164, 186, large)-net in base 4, because
- 20 times m-reduction [i] would yield (164, 166, large)-net in base 4, but