Best Known (14, 18, s)-Nets in Base 5
(14, 18, 1953)-Net over F5 — Constructive and digital
Digital (14, 18, 1953)-net over F5, using
- net defined by OOA [i] based on linear OOA(518, 1953, F5, 4, 4) (dual of [(1953, 4), 7794, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(518, 1953, F5, 3, 4) (dual of [(1953, 3), 5841, 5]-NRT-code), using
(14, 18, 7206)-Net over F5 — Digital
Digital (14, 18, 7206)-net over F5, using
- net defined by OOA [i] based on linear OOA(518, 7206, F5, 4, 4) (dual of [(7206, 4), 28806, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(518, 7206, F5, 3, 4) (dual of [(7206, 3), 21600, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(518, 7206, F5, 4) (dual of [7206, 7188, 5]-code), using
- 4074 step Varšamov–Edel lengthening with (ri) = (1, 1315 times 0, 1, 2757 times 0) [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(516, 3125, F5, 4) (dual of [3125, 3109, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(511, 3125, F5, 3) (dual of [3125, 3114, 4]-code or 3125-cap in PG(10,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 4074 step Varšamov–Edel lengthening with (ri) = (1, 1315 times 0, 1, 2757 times 0) [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(518, 7206, F5, 4) (dual of [7206, 7188, 5]-code), using
- appending kth column [i] based on linear OOA(518, 7206, F5, 3, 4) (dual of [(7206, 3), 21600, 5]-NRT-code), using
(14, 18, 690532)-Net in Base 5 — Upper bound on s
There is no (14, 18, 690533)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 3 814697 641241 > 518 [i]