Best Known (117, 117+22, s)-Nets in Base 8
(117, 117+22, 190653)-Net over F8 — Constructive and digital
Digital (117, 139, 190653)-net over F8, using
- 81 times duplication [i] based on digital (116, 138, 190653)-net over F8, using
- net defined by OOA [i] based on linear OOA(8138, 190653, F8, 22, 22) (dual of [(190653, 22), 4194228, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8138, 2097183, F8, 22) (dual of [2097183, 2097045, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8138, 2097184, F8, 22) (dual of [2097184, 2097046, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(8134, 2097152, F8, 22) (dual of [2097152, 2097018, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8138, 2097184, F8, 22) (dual of [2097184, 2097046, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8138, 2097183, F8, 22) (dual of [2097183, 2097045, 23]-code), using
- net defined by OOA [i] based on linear OOA(8138, 190653, F8, 22, 22) (dual of [(190653, 22), 4194228, 23]-NRT-code), using
(117, 117+22, 2020815)-Net over F8 — Digital
Digital (117, 139, 2020815)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8139, 2020815, F8, 22) (dual of [2020815, 2020676, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 2097185, F8, 22) (dual of [2097185, 2097046, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8138, 2097184, F8, 22) (dual of [2097184, 2097046, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(8134, 2097152, F8, 22) (dual of [2097152, 2097018, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8138, 2097184, F8, 22) (dual of [2097184, 2097046, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 2097185, F8, 22) (dual of [2097185, 2097046, 23]-code), using
(117, 117+22, large)-Net in Base 8 — Upper bound on s
There is no (117, 139, large)-net in base 8, because
- 20 times m-reduction [i] would yield (117, 119, large)-net in base 8, but