Best Known (48−8, 48, s)-Nets in Base 5
(48−8, 48, 19545)-Net over F5 — Constructive and digital
Digital (40, 48, 19545)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (35, 43, 19533)-net over F5, using
- net defined by OOA [i] based on linear OOA(543, 19533, F5, 8, 8) (dual of [(19533, 8), 156221, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(543, 78132, F5, 8) (dual of [78132, 78089, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(543, 78132, F5, 8) (dual of [78132, 78089, 9]-code), using
- net defined by OOA [i] based on linear OOA(543, 19533, F5, 8, 8) (dual of [(19533, 8), 156221, 9]-NRT-code), using
(48−8, 48, 78152)-Net over F5 — Digital
Digital (40, 48, 78152)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(548, 78152, F5, 8) (dual of [78152, 78104, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(3) ⊂ Ce(2) [i] based on
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(515, 78125, F5, 3) (dual of [78125, 78110, 4]-code or 78125-cap in PG(14,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-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
(48−8, 48, large)-Net in Base 5 — Upper bound on s
There is no (40, 48, large)-net in base 5, because
- 6 times m-reduction [i] would yield (40, 42, large)-net in base 5, but