Best Known (163, 163+24, s)-Nets in Base 4
(163, 163+24, 87384)-Net over F4 — Constructive and digital
Digital (163, 187, 87384)-net over F4, using
- 41 times duplication [i] based on digital (162, 186, 87384)-net over F4, using
- t-expansion [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
- t-expansion [i] based on digital (161, 186, 87384)-net over F4, using
(163, 163+24, 524306)-Net over F4 — Digital
Digital (163, 187, 524306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4187, 524306, F4, 2, 24) (dual of [(524306, 2), 1048425, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(4187, 524306, F4, 2, 25) (dual of [(524306, 2), 1048425, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4187, 1048612, F4, 25) (dual of [1048612, 1048425, 26]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(4186, 1048611, F4, 25) (dual of [1048611, 1048425, 26]-code), using
- OOA 2-folding [i] based on linear OA(4187, 1048612, F4, 25) (dual of [1048612, 1048425, 26]-code), using
- strength reduction [i] based on linear OOA(4187, 524306, F4, 2, 25) (dual of [(524306, 2), 1048425, 26]-NRT-code), using
(163, 163+24, large)-Net in Base 4 — Upper bound on s
There is no (163, 187, large)-net in base 4, because
- 22 times m-reduction [i] would yield (163, 165, large)-net in base 4, but