Best Known (67, 67+14, s)-Nets in Base 5
(67, 67+14, 11164)-Net over F5 — Constructive and digital
Digital (67, 81, 11164)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 11164, F5, 14, 14) (dual of [(11164, 14), 156215, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(581, 78148, F5, 14) (dual of [78148, 78067, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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]
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(581, 78148, F5, 14) (dual of [78148, 78067, 15]-code), using
(67, 67+14, 60401)-Net over F5 — Digital
Digital (67, 81, 60401)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 60401, F5, 14) (dual of [60401, 60320, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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]
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 78149, F5, 14) (dual of [78149, 78068, 15]-code), using
(67, 67+14, large)-Net in Base 5 — Upper bound on s
There is no (67, 81, large)-net in base 5, because
- 12 times m-reduction [i] would yield (67, 69, large)-net in base 5, but