Best Known (159, 159+23, s)-Nets in Base 4
(159, 159+23, 95330)-Net over F4 — Constructive and digital
Digital (159, 182, 95330)-net over F4, using
- 41 times duplication [i] based on digital (158, 181, 95330)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 95330, F4, 23, 23) (dual of [(95330, 23), 2192409, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4181, 1048631, F4, 23) (dual of [1048631, 1048450, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 1048636, F4, 23) (dual of [1048636, 1048455, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4181, 1048636, F4, 23) (dual of [1048636, 1048455, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4181, 1048631, F4, 23) (dual of [1048631, 1048450, 24]-code), using
- net defined by OOA [i] based on linear OOA(4181, 95330, F4, 23, 23) (dual of [(95330, 23), 2192409, 24]-NRT-code), using
(159, 159+23, 524318)-Net over F4 — Digital
Digital (159, 182, 524318)-net over F4, using
- 41 times duplication [i] based on digital (158, 181, 524318)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4181, 524318, F4, 2, 23) (dual of [(524318, 2), 1048455, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4181, 1048636, F4, 23) (dual of [1048636, 1048455, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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(22) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4181, 1048636, F4, 23) (dual of [1048636, 1048455, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4181, 524318, F4, 2, 23) (dual of [(524318, 2), 1048455, 24]-NRT-code), using
(159, 159+23, large)-Net in Base 4 — Upper bound on s
There is no (159, 182, large)-net in base 4, because
- 21 times m-reduction [i] would yield (159, 161, large)-net in base 4, but