Best Known (165−28, 165, s)-Nets in Base 8
(165−28, 165, 37449)-Net over F8 — Constructive and digital
Digital (137, 165, 37449)-net over F8, using
- 81 times duplication [i] based on digital (136, 164, 37449)-net over F8, using
- net defined by OOA [i] based on linear OOA(8164, 37449, F8, 28, 28) (dual of [(37449, 28), 1048408, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8164, 524286, F8, 28) (dual of [524286, 524122, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 524288, F8, 28) (dual of [524288, 524124, 29]-code), using
- trace code [i] based on linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- trace code [i] based on linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 524288, F8, 28) (dual of [524288, 524124, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(8164, 524286, F8, 28) (dual of [524286, 524122, 29]-code), using
- net defined by OOA [i] based on linear OOA(8164, 37449, F8, 28, 28) (dual of [(37449, 28), 1048408, 29]-NRT-code), using
(165−28, 165, 524296)-Net over F8 — Digital
Digital (137, 165, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8165, 524296, F8, 28) (dual of [524296, 524131, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8164, 524294, F8, 28) (dual of [524294, 524130, 29]-code), using
- trace code [i] based on linear OA(6482, 262147, F64, 28) (dual of [262147, 262065, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6479, 262144, F64, 27) (dual of [262144, 262065, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- trace code [i] based on linear OA(6482, 262147, F64, 28) (dual of [262147, 262065, 29]-code), using
- linear OA(8164, 524295, F8, 27) (dual of [524295, 524131, 28]-code), using Gilbert–Varšamov bound and bm = 8164 > Vbs−1(k−1) = 1 190175 360518 227145 312742 993634 217906 384889 507672 142228 786028 976818 083995 727836 965157 319194 696140 035952 968738 635573 077380 809873 830043 009799 290880 [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(8164, 524294, F8, 28) (dual of [524294, 524130, 29]-code), using
- construction X with Varšamov bound [i] based on
(165−28, 165, large)-Net in Base 8 — Upper bound on s
There is no (137, 165, large)-net in base 8, because
- 26 times m-reduction [i] would yield (137, 139, large)-net in base 8, but