Best Known (15, 19, s)-Nets in Base 8
(15, 19, 131075)-Net over F8 — Constructive and digital
Digital (15, 19, 131075)-net over F8, using
- net defined by OOA [i] based on linear OOA(819, 131075, F8, 4, 4) (dual of [(131075, 4), 524281, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(819, 131075, F8, 3, 4) (dual of [(131075, 3), 393206, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(819, 262150, F8, 4) (dual of [262150, 262131, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(819, 262144, F8, 4) (dual of [262144, 262125, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(813, 262144, F8, 3) (dual of [262144, 262131, 4]-code or 262144-cap in PG(12,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(819, 262150, F8, 4) (dual of [262150, 262131, 5]-code), using
- appending kth column [i] based on linear OOA(819, 131075, F8, 3, 4) (dual of [(131075, 3), 393206, 5]-NRT-code), using
(15, 19, 262150)-Net over F8 — Digital
Digital (15, 19, 262150)-net over F8, using
- net defined by OOA [i] based on linear OOA(819, 262150, F8, 4, 4) (dual of [(262150, 4), 1048581, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(819, 262150, F8, 3, 4) (dual of [(262150, 3), 786431, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(819, 262150, F8, 4) (dual of [262150, 262131, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(819, 262144, F8, 4) (dual of [262144, 262125, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(813, 262144, F8, 3) (dual of [262144, 262131, 4]-code or 262144-cap in PG(12,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(819, 262150, F8, 4) (dual of [262150, 262131, 5]-code), using
- appending kth column [i] based on linear OOA(819, 262150, F8, 3, 4) (dual of [(262150, 3), 786431, 5]-NRT-code), using
(15, 19, large)-Net in Base 8 — Upper bound on s
There is no (15, 19, large)-net in base 8, because
- 2 times m-reduction [i] would yield (15, 17, large)-net in base 8, but