Best Known (9, 9+6, s)-Nets in Base 5
(9, 9+6, 66)-Net over F5 — Constructive and digital
Digital (9, 15, 66)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 66)-net over F5, using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(55, 66, F5, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- digital (4, 10, 33)-net over F5, using
- digital (2, 5, 66)-net over F5, using
(9, 9+6, 132)-Net over F5 — Digital
Digital (9, 15, 132)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(515, 132, F5, 6) (dual of [132, 117, 7]-code), using
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(514, 129, F5, 6) (dual of [129, 115, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(513, 125, F5, 6) (dual of [125, 112, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(510, 125, F5, 4) (dual of [125, 115, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(514, 129, F5, 6) (dual of [129, 115, 7]-code), using
(9, 9+6, 1417)-Net in Base 5 — Upper bound on s
There is no (9, 15, 1418)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 30525 552585 > 515 [i]