Best Known (157−23, 157, s)-Nets in Base 4
(157−23, 157, 23833)-Net over F4 — Constructive and digital
Digital (134, 157, 23833)-net over F4, using
- 41 times duplication [i] based on digital (133, 156, 23833)-net over F4, using
- net defined by OOA [i] based on linear OOA(4156, 23833, F4, 23, 23) (dual of [(23833, 23), 548003, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4156, 262164, F4, 23) (dual of [262164, 262008, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4155, 262163, F4, 23) (dual of [262163, 262008, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [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(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4155, 262163, F4, 23) (dual of [262163, 262008, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4156, 262164, F4, 23) (dual of [262164, 262008, 24]-code), using
- net defined by OOA [i] based on linear OOA(4156, 23833, F4, 23, 23) (dual of [(23833, 23), 548003, 24]-NRT-code), using
(157−23, 157, 128262)-Net over F4 — Digital
Digital (134, 157, 128262)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4157, 128262, F4, 2, 23) (dual of [(128262, 2), 256367, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 131082, F4, 2, 23) (dual of [(131082, 2), 262007, 24]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4156, 131082, F4, 2, 23) (dual of [(131082, 2), 262008, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4156, 262164, F4, 23) (dual of [262164, 262008, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4155, 262163, F4, 23) (dual of [262163, 262008, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [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(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4155, 262163, F4, 23) (dual of [262163, 262008, 24]-code), using
- OOA 2-folding [i] based on linear OA(4156, 262164, F4, 23) (dual of [262164, 262008, 24]-code), using
- 41 times duplication [i] based on linear OOA(4156, 131082, F4, 2, 23) (dual of [(131082, 2), 262008, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4157, 131082, F4, 2, 23) (dual of [(131082, 2), 262007, 24]-NRT-code), using
(157−23, 157, large)-Net in Base 4 — Upper bound on s
There is no (134, 157, large)-net in base 4, because
- 21 times m-reduction [i] would yield (134, 136, large)-net in base 4, but