Best Known (34−6, 34, s)-Nets in Base 5
(34−6, 34, 130211)-Net over F5 — Constructive and digital
Digital (28, 34, 130211)-net over F5, using
- net defined by OOA [i] based on linear OOA(534, 130211, F5, 6, 6) (dual of [(130211, 6), 781232, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(534, 390633, F5, 6) (dual of [390633, 390599, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(534, 390634, F5, 6) (dual of [390634, 390600, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(525, 390625, F5, 4) (dual of [390625, 390600, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(534, 390634, F5, 6) (dual of [390634, 390600, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(534, 390633, F5, 6) (dual of [390633, 390599, 7]-code), using
(34−6, 34, 323216)-Net over F5 — Digital
Digital (28, 34, 323216)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(534, 323216, F5, 6) (dual of [323216, 323182, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(534, 390634, F5, 6) (dual of [390634, 390600, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(525, 390625, F5, 4) (dual of [390625, 390600, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(534, 390634, F5, 6) (dual of [390634, 390600, 7]-code), using
(34−6, 34, large)-Net in Base 5 — Upper bound on s
There is no (28, 34, large)-net in base 5, because
- 4 times m-reduction [i] would yield (28, 30, large)-net in base 5, but