Best Known (137−25, 137, s)-Nets in Base 8
(137−25, 137, 21848)-Net over F8 — Constructive and digital
Digital (112, 137, 21848)-net over F8, using
- 83 times duplication [i] based on digital (109, 134, 21848)-net over F8, using
- net defined by OOA [i] based on linear OOA(8134, 21848, F8, 25, 25) (dual of [(21848, 25), 546066, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8134, 262177, F8, 25) (dual of [262177, 262043, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8134, 262181, F8, 25) (dual of [262181, 262047, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- 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(897, 262144, F8, 19) (dual of [262144, 262047, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 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(24) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8134, 262181, F8, 25) (dual of [262181, 262047, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8134, 262177, F8, 25) (dual of [262177, 262043, 26]-code), using
- net defined by OOA [i] based on linear OOA(8134, 21848, F8, 25, 25) (dual of [(21848, 25), 546066, 26]-NRT-code), using
(137−25, 137, 262190)-Net over F8 — Digital
Digital (112, 137, 262190)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8137, 262190, F8, 25) (dual of [262190, 262053, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- 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(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(810, 46, F8, 6) (dual of [46, 36, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
(137−25, 137, large)-Net in Base 8 — Upper bound on s
There is no (112, 137, large)-net in base 8, because
- 23 times m-reduction [i] would yield (112, 114, large)-net in base 8, but