Best Known (48, 48+12, s)-Nets in Base 5
(48, 48+12, 2608)-Net over F5 — Constructive and digital
Digital (48, 60, 2608)-net over F5, using
- net defined by OOA [i] based on linear OOA(560, 2608, F5, 12, 12) (dual of [(2608, 12), 31236, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(560, 15648, F5, 12) (dual of [15648, 15588, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 15649, F5, 12) (dual of [15649, 15589, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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(11) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(560, 15649, F5, 12) (dual of [15649, 15589, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(560, 15648, F5, 12) (dual of [15648, 15588, 13]-code), using
(48, 48+12, 15054)-Net over F5 — Digital
Digital (48, 60, 15054)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(560, 15054, F5, 12) (dual of [15054, 14994, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 15649, F5, 12) (dual of [15649, 15589, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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(11) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(560, 15649, F5, 12) (dual of [15649, 15589, 13]-code), using
(48, 48+12, 7309066)-Net in Base 5 — Upper bound on s
There is no (48, 60, 7309067)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 867362 287428 962045 598046 068390 372889 486313 > 560 [i]