Best Known (135, 135+23, s)-Nets in Base 4
(135, 135+23, 23833)-Net over F4 — Constructive and digital
Digital (135, 158, 23833)-net over F4, using
- 42 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
(135, 135+23, 131084)-Net over F4 — Digital
Digital (135, 158, 131084)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4158, 131084, F4, 2, 23) (dual of [(131084, 2), 262010, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4158, 262168, F4, 23) (dual of [262168, 262010, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [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(4127, 262144, F4, 19) (dual of [262144, 262017, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(4158, 262168, F4, 23) (dual of [262168, 262010, 24]-code), using
(135, 135+23, large)-Net in Base 4 — Upper bound on s
There is no (135, 158, large)-net in base 4, because
- 21 times m-reduction [i] would yield (135, 137, large)-net in base 4, but