Best Known (38, 45, s)-Nets in Base 5
(38, 45, 130236)-Net over F5 — Constructive and digital
Digital (38, 45, 130236)-net over F5, using
- net defined by OOA [i] based on linear OOA(545, 130236, F5, 9, 7) (dual of [(130236, 9), 1172079, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(545, 130237, F5, 3, 7) (dual of [(130237, 3), 390666, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(54, 26, F5, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
- linear OOA(541, 130211, F5, 3, 7) (dual of [(130211, 3), 390592, 8]-NRT-code), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on linear OA(541, 390633, F5, 7) (dual of [390633, 390592, 8]-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(545, 130237, F5, 3, 7) (dual of [(130237, 3), 390666, 8]-NRT-code), using
(38, 45, 390659)-Net over F5 — Digital
Digital (38, 45, 390659)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(545, 390659, F5, 7) (dual of [390659, 390614, 8]-code), using
- (u, u+v)-construction [i] based on
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- 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
- (u, u+v)-construction [i] based on
(38, 45, large)-Net in Base 5 — Upper bound on s
There is no (38, 45, large)-net in base 5, because
- 5 times m-reduction [i] would yield (38, 40, large)-net in base 5, but