Best Known (34, 42, s)-Nets in Base 5
(34, 42, 3919)-Net over F5 — Constructive and digital
Digital (34, 42, 3919)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (29, 37, 3907)-net over F5, using
- net defined by OOA [i] based on linear OOA(537, 3907, F5, 8, 8) (dual of [(3907, 8), 31219, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(537, 15628, F5, 8) (dual of [15628, 15591, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(537, 15631, F5, 8) (dual of [15631, 15594, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [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(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(537, 15631, F5, 8) (dual of [15631, 15594, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(537, 15628, F5, 8) (dual of [15628, 15591, 9]-code), using
- net defined by OOA [i] based on linear OOA(537, 3907, F5, 8, 8) (dual of [(3907, 8), 31219, 9]-NRT-code), using
(34, 42, 15649)-Net over F5 — Digital
Digital (34, 42, 15649)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(542, 15649, F5, 8) (dual of [15649, 15607, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [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(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(513, 15625, F5, 3) (dual of [15625, 15612, 4]-code or 15625-cap in PG(12,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
(34, 42, large)-Net in Base 5 — Upper bound on s
There is no (34, 42, large)-net in base 5, because
- 6 times m-reduction [i] would yield (34, 36, large)-net in base 5, but