Best Known (37, 37+7, s)-Nets in Base 5
(37, 37+7, 130216)-Net over F5 — Constructive and digital
Digital (37, 44, 130216)-net over F5, using
- net defined by OOA [i] based on linear OOA(544, 130216, F5, 9, 7) (dual of [(130216, 9), 1171900, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(544, 130217, F5, 3, 7) (dual of [(130217, 3), 390607, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(53, 6, F5, 3, 3) (dual of [(6, 3), 15, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;15,5) [i]
- 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
- linear OOA(53, 6, F5, 3, 3) (dual of [(6, 3), 15, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(544, 130217, F5, 3, 7) (dual of [(130217, 3), 390607, 8]-NRT-code), using
(37, 37+7, 390644)-Net over F5 — Digital
Digital (37, 44, 390644)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(544, 390644, F5, 7) (dual of [390644, 390600, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [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(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(53, 19, F5, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
(37, 37+7, large)-Net in Base 5 — Upper bound on s
There is no (37, 44, large)-net in base 5, because
- 5 times m-reduction [i] would yield (37, 39, large)-net in base 5, but