Best Known (34, 34+7, s)-Nets in Base 5
(34, 34+7, 130210)-Net over F5 — Constructive and digital
Digital (34, 41, 130210)-net over F5, using
- net defined by OOA [i] based on linear OOA(541, 130210, F5, 7, 7) (dual of [(130210, 7), 911429, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(541, 390631, F5, 7) (dual of [390631, 390590, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(541, 390633, F5, 7) (dual of [390633, 390592, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(541, 390633, F5, 7) (dual of [390633, 390592, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(541, 390631, F5, 7) (dual of [390631, 390590, 8]-code), using
(34, 34+7, 254408)-Net over F5 — Digital
Digital (34, 41, 254408)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(541, 254408, F5, 7) (dual of [254408, 254367, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using
(34, 34+7, large)-Net in Base 5 — Upper bound on s
There is no (34, 41, large)-net in base 5, because
- 5 times m-reduction [i] would yield (34, 36, large)-net in base 5, but