Best Known (12, 16, s)-Nets in Base 5
(12, 16, 1565)-Net over F5 — Constructive and digital
Digital (12, 16, 1565)-net over F5, using
- net defined by OOA [i] based on linear OOA(516, 1565, F5, 4, 4) (dual of [(1565, 4), 6244, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(516, 1565, F5, 3, 4) (dual of [(1565, 3), 4679, 5]-NRT-code), using
- OA 2-folding and stacking [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
- OA 2-folding and stacking [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- appending kth column [i] based on linear OOA(516, 1565, F5, 3, 4) (dual of [(1565, 3), 4679, 5]-NRT-code), using
(12, 16, 3130)-Net over F5 — Digital
Digital (12, 16, 3130)-net over F5, using
- net defined by OOA [i] based on linear OOA(516, 3130, F5, 4, 4) (dual of [(3130, 4), 12504, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(516, 3130, F5, 3, 4) (dual of [(3130, 3), 9374, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(516, 3130, F5, 4) (dual of [3130, 3114, 5]-code), using
- appending kth column [i] based on linear OOA(516, 3130, F5, 3, 4) (dual of [(3130, 3), 9374, 5]-NRT-code), using
(12, 16, 138105)-Net in Base 5 — Upper bound on s
There is no (12, 16, 138106)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 152588 347585 > 516 [i]