Best Known (75−18, 75, s)-Nets in Base 5
(75−18, 75, 349)-Net over F5 — Constructive and digital
Digital (57, 75, 349)-net over F5, using
- net defined by OOA [i] based on linear OOA(575, 349, F5, 18, 18) (dual of [(349, 18), 6207, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(575, 3141, F5, 18) (dual of [3141, 3066, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(575, 3141, F5, 18) (dual of [3141, 3066, 19]-code), using
(75−18, 75, 2895)-Net over F5 — Digital
Digital (57, 75, 2895)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(575, 2895, F5, 18) (dual of [2895, 2820, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(575, 3144, F5, 18) (dual of [3144, 3069, 19]-code), using
(75−18, 75, 692528)-Net in Base 5 — Upper bound on s
There is no (57, 75, 692529)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 26469 994051 278332 266843 443343 916065 725562 608924 748325 > 575 [i]