Best Known (54, 66, s)-Nets in Base 8
(54, 66, 43694)-Net over F8 — Constructive and digital
Digital (54, 66, 43694)-net over F8, using
- 82 times duplication [i] based on digital (52, 64, 43694)-net over F8, using
- net defined by OOA [i] based on linear OOA(864, 43694, F8, 12, 12) (dual of [(43694, 12), 524264, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(864, 262164, F8, 12) (dual of [262164, 262100, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 262165, F8, 12) (dual of [262165, 262101, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(83, 21, F8, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(864, 262165, F8, 12) (dual of [262165, 262101, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(864, 262164, F8, 12) (dual of [262164, 262100, 13]-code), using
- net defined by OOA [i] based on linear OOA(864, 43694, F8, 12, 12) (dual of [(43694, 12), 524264, 13]-NRT-code), using
(54, 66, 262169)-Net over F8 — Digital
Digital (54, 66, 262169)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(866, 262169, F8, 12) (dual of [262169, 262103, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(864, 262165, F8, 12) (dual of [262165, 262101, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(83, 21, F8, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(864, 262167, F8, 11) (dual of [262167, 262103, 12]-code), using Gilbert–Varšamov bound and bm = 864 > Vbs−1(k−1) = 119 373758 057281 616058 770564 429015 909365 591642 153502 215628 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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(864, 262165, F8, 12) (dual of [262165, 262101, 13]-code), using
- construction X with Varšamov bound [i] based on
(54, 66, large)-Net in Base 8 — Upper bound on s
There is no (54, 66, large)-net in base 8, because
- 10 times m-reduction [i] would yield (54, 56, large)-net in base 8, but