Best Known (75, 89, s)-Nets in Base 5
(75, 89, 55804)-Net over F5 — Constructive and digital
Digital (75, 89, 55804)-net over F5, using
- net defined by OOA [i] based on linear OOA(589, 55804, F5, 14, 14) (dual of [(55804, 14), 781167, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(589, 390628, F5, 14) (dual of [390628, 390539, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(589, 390628, F5, 14) (dual of [390628, 390539, 15]-code), using
(75, 89, 195316)-Net over F5 — Digital
Digital (75, 89, 195316)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(589, 195316, F5, 2, 14) (dual of [(195316, 2), 390543, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(589, 390632, F5, 14) (dual of [390632, 390543, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- OOA 2-folding [i] based on linear OA(589, 390632, F5, 14) (dual of [390632, 390543, 15]-code), using
(75, 89, large)-Net in Base 5 — Upper bound on s
There is no (75, 89, large)-net in base 5, because
- 12 times m-reduction [i] would yield (75, 77, large)-net in base 5, but