Best Known (40−6, 40, s)-Nets in Base 5
(40−6, 40, 651048)-Net over F5 — Constructive and digital
Digital (34, 40, 651048)-net over F5, using
- net defined by OOA [i] based on linear OOA(540, 651048, F5, 6, 6) (dual of [(651048, 6), 3906248, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(540, 1953144, F5, 6) (dual of [1953144, 1953104, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(540, 1953146, F5, 6) (dual of [1953146, 1953106, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(519, 1953125, F5, 3) (dual of [1953125, 1953106, 4]-code or 1953125-cap in PG(18,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 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(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(540, 1953146, F5, 6) (dual of [1953146, 1953106, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(540, 1953144, F5, 6) (dual of [1953144, 1953104, 7]-code), using
(40−6, 40, 1953146)-Net over F5 — Digital
Digital (34, 40, 1953146)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(540, 1953146, F5, 6) (dual of [1953146, 1953106, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(519, 1953125, F5, 3) (dual of [1953125, 1953106, 4]-code or 1953125-cap in PG(18,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 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(5) ⊂ Ce(2) [i] based on
(40−6, 40, large)-Net in Base 5 — Upper bound on s
There is no (34, 40, large)-net in base 5, because
- 4 times m-reduction [i] would yield (34, 36, large)-net in base 5, but