Best Known (44−8, 44, s)-Nets in Base 5
(44−8, 44, 19535)-Net over F5 — Constructive and digital
Digital (36, 44, 19535)-net over F5, using
- net defined by OOA [i] based on linear OOA(544, 19535, F5, 8, 8) (dual of [(19535, 8), 156236, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(544, 78140, F5, 8) (dual of [78140, 78096, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(544, 78140, F5, 8) (dual of [78140, 78096, 9]-code), using
(44−8, 44, 76459)-Net over F5 — Digital
Digital (36, 44, 76459)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(544, 76459, F5, 8) (dual of [76459, 76415, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(544, 78140, F5, 8) (dual of [78140, 78096, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(544, 78140, F5, 8) (dual of [78140, 78096, 9]-code), using
(44−8, 44, large)-Net in Base 5 — Upper bound on s
There is no (36, 44, large)-net in base 5, because
- 6 times m-reduction [i] would yield (36, 38, large)-net in base 5, but