Best Known (24, 24+7, s)-Nets in Base 5
(24, 24+7, 5210)-Net over F5 — Constructive and digital
Digital (24, 31, 5210)-net over F5, using
- net defined by OOA [i] based on linear OOA(531, 5210, F5, 7, 7) (dual of [(5210, 7), 36439, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(531, 15631, F5, 7) (dual of [15631, 15600, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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 and stacking with additional row [i] based on linear OA(531, 15631, F5, 7) (dual of [15631, 15600, 8]-code), using
(24, 24+7, 10174)-Net over F5 — Digital
Digital (24, 31, 10174)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(531, 10174, F5, 7) (dual of [10174, 10143, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(531, 15624, F5, 7) (dual of [15624, 15593, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(531, 15624, F5, 7) (dual of [15624, 15593, 8]-code), using
(24, 24+7, 4436327)-Net in Base 5 — Upper bound on s
There is no (24, 31, 4436328)-net in base 5, because
- 1 times m-reduction [i] would yield (24, 30, 4436328)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 931 322686 501686 972065 > 530 [i]