Best Known (18−6, 18, s)-Nets in Base 25
(18−6, 18, 5212)-Net over F25 — Constructive and digital
Digital (12, 18, 5212)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 5212, F25, 6, 6) (dual of [(5212, 6), 31254, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2518, 15636, F25, 6) (dual of [15636, 15618, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(2516, 15625, F25, 6) (dual of [15625, 15609, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(257, 15625, F25, 3) (dual of [15625, 15618, 4]-code or 15625-cap in PG(6,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2518, 15636, F25, 6) (dual of [15636, 15618, 7]-code), using
(18−6, 18, 15636)-Net over F25 — Digital
Digital (12, 18, 15636)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2518, 15636, F25, 6) (dual of [15636, 15618, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(2516, 15625, F25, 6) (dual of [15625, 15609, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(257, 15625, F25, 3) (dual of [15625, 15618, 4]-code or 15625-cap in PG(6,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(18−6, 18, large)-Net in Base 25 — Upper bound on s
There is no (12, 18, large)-net in base 25, because
- 4 times m-reduction [i] would yield (12, 14, large)-net in base 25, but