Best Known (38−8, 38, s)-Nets in Base 5
(38−8, 38, 3909)-Net over F5 — Constructive and digital
Digital (30, 38, 3909)-net over F5, using
- net defined by OOA [i] based on linear OOA(538, 3909, F5, 8, 8) (dual of [(3909, 8), 31234, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(538, 15636, F5, 8) (dual of [15636, 15598, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(538, 15638, F5, 8) (dual of [15638, 15600, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 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(538, 15638, F5, 8) (dual of [15638, 15600, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(538, 15636, F5, 8) (dual of [15636, 15598, 9]-code), using
(38−8, 38, 15289)-Net over F5 — Digital
Digital (30, 38, 15289)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(538, 15289, F5, 8) (dual of [15289, 15251, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(538, 15638, F5, 8) (dual of [15638, 15600, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 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(538, 15638, F5, 8) (dual of [15638, 15600, 9]-code), using
(38−8, 38, 2416614)-Net in Base 5 — Upper bound on s
There is no (30, 38, 2416615)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 363 798067 754772 523564 457201 > 538 [i]