Best Known (50−7, 50, s)-Nets in Base 5
(50−7, 50, 651070)-Net over F5 — Constructive and digital
Digital (43, 50, 651070)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 26)-net over F5, using
- net defined by OOA [i] based on linear OOA(54, 26, F5, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
- digital (39, 46, 651044)-net over F5, using
- net defined by OOA [i] based on linear OOA(546, 651044, F5, 7, 7) (dual of [(651044, 7), 4557262, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(546, 1953133, F5, 7) (dual of [1953133, 1953087, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(546, 1953134, F5, 7) (dual of [1953134, 1953088, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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
- discarding factors / shortening the dual code based on linear OA(546, 1953134, F5, 7) (dual of [1953134, 1953088, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(546, 1953133, F5, 7) (dual of [1953133, 1953087, 8]-code), using
- net defined by OOA [i] based on linear OOA(546, 651044, F5, 7, 7) (dual of [(651044, 7), 4557262, 8]-NRT-code), using
- digital (1, 4, 26)-net over F5, using
(50−7, 50, 1953160)-Net over F5 — Digital
Digital (43, 50, 1953160)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(550, 1953160, F5, 7) (dual of [1953160, 1953110, 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(546, 1953134, F5, 7) (dual of [1953134, 1953088, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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
(50−7, 50, large)-Net in Base 5 — Upper bound on s
There is no (43, 50, large)-net in base 5, because
- 5 times m-reduction [i] would yield (43, 45, large)-net in base 5, but