Best Known (85−12, 85, s)-Nets in Base 5
(85−12, 85, 325524)-Net over F5 — Constructive and digital
Digital (73, 85, 325524)-net over F5, using
- net defined by OOA [i] based on linear OOA(585, 325524, F5, 12, 12) (dual of [(325524, 12), 3906203, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(585, 1953144, F5, 12) (dual of [1953144, 1953059, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 1953146, F5, 12) (dual of [1953146, 1953061, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(585, 1953146, F5, 12) (dual of [1953146, 1953061, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(585, 1953144, F5, 12) (dual of [1953144, 1953059, 13]-code), using
(85−12, 85, 976573)-Net over F5 — Digital
Digital (73, 85, 976573)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(585, 976573, F5, 2, 12) (dual of [(976573, 2), 1953061, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(585, 1953146, F5, 12) (dual of [1953146, 1953061, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(585, 1953146, F5, 12) (dual of [1953146, 1953061, 13]-code), using
(85−12, 85, large)-Net in Base 5 — Upper bound on s
There is no (73, 85, large)-net in base 5, because
- 10 times m-reduction [i] would yield (73, 75, large)-net in base 5, but