Best Known (163−23, 163, s)-Nets in Base 4
(163−23, 163, 23835)-Net over F4 — Constructive and digital
Digital (140, 163, 23835)-net over F4, using
- 42 times duplication [i] based on digital (138, 161, 23835)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 23835, F4, 23, 23) (dual of [(23835, 23), 548044, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4161, 262186, F4, 23) (dual of [262186, 262025, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 262187, F4, 23) (dual of [262187, 262026, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4161, 262187, F4, 23) (dual of [262187, 262026, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4161, 262186, F4, 23) (dual of [262186, 262025, 24]-code), using
- net defined by OOA [i] based on linear OOA(4161, 23835, F4, 23, 23) (dual of [(23835, 23), 548044, 24]-NRT-code), using
(163−23, 163, 131097)-Net over F4 — Digital
Digital (140, 163, 131097)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4163, 131097, F4, 2, 23) (dual of [(131097, 2), 262031, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4163, 262194, F4, 23) (dual of [262194, 262031, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 262195, F4, 23) (dual of [262195, 262032, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4163, 262195, F4, 23) (dual of [262195, 262032, 24]-code), using
- OOA 2-folding [i] based on linear OA(4163, 262194, F4, 23) (dual of [262194, 262031, 24]-code), using
(163−23, 163, large)-Net in Base 4 — Upper bound on s
There is no (140, 163, large)-net in base 4, because
- 21 times m-reduction [i] would yield (140, 142, large)-net in base 4, but