Best Known (39−7, 39, s)-Nets in Base 5
(39−7, 39, 26049)-Net over F5 — Constructive and digital
Digital (32, 39, 26049)-net over F5, using
- net defined by OOA [i] based on linear OOA(539, 26049, F5, 9, 7) (dual of [(26049, 9), 234402, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(539, 26050, F5, 3, 7) (dual of [(26050, 3), 78111, 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(536, 26044, F5, 3, 7) (dual of [(26044, 3), 78096, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(536, 78132, F5, 7) (dual of [78132, 78096, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(536, 78132, F5, 7) (dual of [78132, 78096, 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(539, 26050, F5, 3, 7) (dual of [(26050, 3), 78111, 8]-NRT-code), using
(39−7, 39, 78142)-Net over F5 — Digital
Digital (32, 39, 78142)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(539, 78142, F5, 7) (dual of [78142, 78103, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(522, 78125, F5, 4) (dual of [78125, 78103, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(53, 17, F5, 2) (dual of [17, 14, 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
(39−7, 39, large)-Net in Base 5 — Upper bound on s
There is no (32, 39, large)-net in base 5, because
- 5 times m-reduction [i] would yield (32, 34, large)-net in base 5, but