Best Known (144, 144+27, s)-Nets in Base 4
(144, 144+27, 5045)-Net over F4 — Constructive and digital
Digital (144, 171, 5045)-net over F4, using
- net defined by OOA [i] based on linear OOA(4171, 5045, F4, 27, 27) (dual of [(5045, 27), 136044, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 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 Ce(26) ⊂ Ce(20) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
(144, 144+27, 42103)-Net over F4 — Digital
Digital (144, 171, 42103)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4171, 42103, F4, 27) (dual of [42103, 41932, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 65579, F4, 27) (dual of [65579, 65408, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4129, 65537, F4, 21) (dual of [65537, 65408, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 65579, F4, 27) (dual of [65579, 65408, 28]-code), using
(144, 144+27, large)-Net in Base 4 — Upper bound on s
There is no (144, 171, large)-net in base 4, because
- 25 times m-reduction [i] would yield (144, 146, large)-net in base 4, but