Best Known (162, 162+25, s)-Nets in Base 4
(162, 162+25, 87384)-Net over F4 — Constructive and digital
Digital (162, 187, 87384)-net over F4, using
- 41 times duplication [i] based on digital (161, 186, 87384)-net over F4, using
- net defined by OOA [i] based on linear OOA(4186, 87384, F4, 25, 25) (dual of [(87384, 25), 2184414, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4186, 1048609, F4, 25) (dual of [1048609, 1048423, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 1048611, F4, 25) (dual of [1048611, 1048425, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4186, 1048611, F4, 25) (dual of [1048611, 1048425, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4186, 1048609, F4, 25) (dual of [1048609, 1048423, 26]-code), using
- net defined by OOA [i] based on linear OOA(4186, 87384, F4, 25, 25) (dual of [(87384, 25), 2184414, 26]-NRT-code), using
(162, 162+25, 349537)-Net over F4 — Digital
Digital (162, 187, 349537)-net over F4, using
- 41 times duplication [i] based on digital (161, 186, 349537)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4186, 349537, F4, 3, 25) (dual of [(349537, 3), 1048425, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4186, 1048611, F4, 25) (dual of [1048611, 1048425, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- OOA 3-folding [i] based on linear OA(4186, 1048611, F4, 25) (dual of [1048611, 1048425, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4186, 349537, F4, 3, 25) (dual of [(349537, 3), 1048425, 26]-NRT-code), using
(162, 162+25, large)-Net in Base 4 — Upper bound on s
There is no (162, 187, large)-net in base 4, because
- 23 times m-reduction [i] would yield (162, 164, large)-net in base 4, but