Best Known (52, 69, s)-Nets in Base 5
(52, 69, 392)-Net over F5 — Constructive and digital
Digital (52, 69, 392)-net over F5, using
- net defined by OOA [i] based on linear OOA(569, 392, F5, 17, 17) (dual of [(392, 17), 6595, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(569, 3137, F5, 17) (dual of [3137, 3068, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(569, 3138, F5, 17) (dual of [3138, 3069, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(53, 13, F5, 2) (dual of [13, 10, 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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(569, 3138, F5, 17) (dual of [3138, 3069, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(569, 3137, F5, 17) (dual of [3137, 3068, 18]-code), using
(52, 69, 2358)-Net over F5 — Digital
Digital (52, 69, 2358)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 2358, F5, 17) (dual of [2358, 2289, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(569, 3138, F5, 17) (dual of [3138, 3069, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(53, 13, F5, 2) (dual of [13, 10, 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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(569, 3138, F5, 17) (dual of [3138, 3069, 18]-code), using
(52, 69, 822000)-Net in Base 5 — Upper bound on s
There is no (52, 69, 822001)-net in base 5, because
- 1 times m-reduction [i] would yield (52, 68, 822001)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 338814 057431 007445 604705 266965 354132 186699 526625 > 568 [i]