Best Known (44, 56, s)-Nets in Base 8
(44, 56, 5464)-Net over F8 — Constructive and digital
Digital (44, 56, 5464)-net over F8, using
- 82 times duplication [i] based on digital (42, 54, 5464)-net over F8, using
- net defined by OOA [i] based on linear OOA(854, 5464, F8, 12, 12) (dual of [(5464, 12), 65514, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(854, 32784, F8, 12) (dual of [32784, 32730, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(854, 32786, F8, 12) (dual of [32786, 32732, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 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(854, 32786, F8, 12) (dual of [32786, 32732, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(854, 32784, F8, 12) (dual of [32784, 32730, 13]-code), using
- net defined by OOA [i] based on linear OOA(854, 5464, F8, 12, 12) (dual of [(5464, 12), 65514, 13]-NRT-code), using
(44, 56, 32790)-Net over F8 — Digital
Digital (44, 56, 32790)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 32790, F8, 12) (dual of [32790, 32734, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(854, 32786, F8, 12) (dual of [32786, 32732, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 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(854, 32788, F8, 11) (dual of [32788, 32734, 12]-code), using Gilbert–Varšamov bound and bm = 854 > Vbs−1(k−1) = 111598 145499 111330 310628 671878 120389 732553 061214 [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(854, 32786, F8, 12) (dual of [32786, 32732, 13]-code), using
- construction X with Varšamov bound [i] based on
(44, 56, large)-Net in Base 8 — Upper bound on s
There is no (44, 56, large)-net in base 8, because
- 10 times m-reduction [i] would yield (44, 46, large)-net in base 8, but