Best Known (129−22, 129, s)-Nets in Base 8
(129−22, 129, 47663)-Net over F8 — Constructive and digital
Digital (107, 129, 47663)-net over F8, using
- 81 times duplication [i] based on digital (106, 128, 47663)-net over F8, using
- net defined by OOA [i] based on linear OOA(8128, 47663, F8, 22, 22) (dual of [(47663, 22), 1048458, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8128, 524293, F8, 22) (dual of [524293, 524165, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 524294, F8, 22) (dual of [524294, 524166, 23]-code), using
- trace code [i] based on linear OA(6464, 262147, F64, 22) (dual of [262147, 262083, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(6464, 262147, F64, 22) (dual of [262147, 262083, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 524294, F8, 22) (dual of [524294, 524166, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8128, 524293, F8, 22) (dual of [524293, 524165, 23]-code), using
- net defined by OOA [i] based on linear OOA(8128, 47663, F8, 22, 22) (dual of [(47663, 22), 1048458, 23]-NRT-code), using
(129−22, 129, 524296)-Net over F8 — Digital
Digital (107, 129, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 524296, F8, 22) (dual of [524296, 524167, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8128, 524294, F8, 22) (dual of [524294, 524166, 23]-code), using
- trace code [i] based on linear OA(6464, 262147, F64, 22) (dual of [262147, 262083, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(6464, 262147, F64, 22) (dual of [262147, 262083, 23]-code), using
- linear OA(8128, 524295, F8, 21) (dual of [524295, 524167, 22]-code), using Gilbert–Varšamov bound and bm = 8128 > Vbs−1(k−1) = 80756 765707 905955 968913 020211 700402 436977 335646 525220 003901 038801 501594 028978 977775 255019 115890 892126 217631 137792 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8128, 524294, F8, 22) (dual of [524294, 524166, 23]-code), using
- construction X with Varšamov bound [i] based on
(129−22, 129, large)-Net in Base 8 — Upper bound on s
There is no (107, 129, large)-net in base 8, because
- 20 times m-reduction [i] would yield (107, 109, large)-net in base 8, but