Best Known (19, 26, s)-Nets in Base 8
(19, 26, 2731)-Net over F8 — Constructive and digital
Digital (19, 26, 2731)-net over F8, using
- net defined by OOA [i] based on linear OOA(826, 2731, F8, 7, 7) (dual of [(2731, 7), 19091, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(826, 8194, F8, 7) (dual of [8194, 8168, 8]-code), using
- trace code [i] based on linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- trace code [i] based on linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(826, 8194, F8, 7) (dual of [8194, 8168, 8]-code), using
(19, 26, 8196)-Net over F8 — Digital
Digital (19, 26, 8196)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
(19, 26, large)-Net in Base 8 — Upper bound on s
There is no (19, 26, large)-net in base 8, because
- 5 times m-reduction [i] would yield (19, 21, large)-net in base 8, but