Best Known (171−31, 171, s)-Nets in Base 8
(171−31, 171, 17479)-Net over F8 — Constructive and digital
Digital (140, 171, 17479)-net over F8, using
- 81 times duplication [i] based on digital (139, 170, 17479)-net over F8, using
- net defined by OOA [i] based on linear OOA(8170, 17479, F8, 31, 31) (dual of [(17479, 31), 541679, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8170, 262186, F8, 31) (dual of [262186, 262016, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 262187, F8, 31) (dual of [262187, 262017, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8127, 262144, F8, 25) (dual of [262144, 262017, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(87, 43, F8, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 262187, F8, 31) (dual of [262187, 262017, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8170, 262186, F8, 31) (dual of [262186, 262016, 32]-code), using
- net defined by OOA [i] based on linear OOA(8170, 17479, F8, 31, 31) (dual of [(17479, 31), 541679, 32]-NRT-code), using
(171−31, 171, 262189)-Net over F8 — Digital
Digital (140, 171, 262189)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8171, 262189, F8, 31) (dual of [262189, 262018, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8170, 262187, F8, 31) (dual of [262187, 262017, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8127, 262144, F8, 25) (dual of [262144, 262017, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(87, 43, F8, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(8170, 262188, F8, 30) (dual of [262188, 262018, 31]-code), using Gilbert–Varšamov bound and bm = 8170 > Vbs−1(k−1) = 5 016053 302511 758921 702350 391057 937969 847287 785423 947670 100842 644658 467119 400216 906883 982337 068539 506507 935193 883039 772859 548614 666015 358020 869510 772912 [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(8170, 262187, F8, 31) (dual of [262187, 262017, 32]-code), using
- construction X with Varšamov bound [i] based on
(171−31, 171, large)-Net in Base 8 — Upper bound on s
There is no (140, 171, large)-net in base 8, because
- 29 times m-reduction [i] would yield (140, 142, large)-net in base 8, but