Best Known (72, 72+13, s)-Nets in Base 5
(72, 72+13, 65108)-Net over F5 — Constructive and digital
Digital (72, 85, 65108)-net over F5, using
- net defined by OOA [i] based on linear OOA(585, 65108, F5, 13, 13) (dual of [(65108, 13), 846319, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(585, 390649, F5, 13) (dual of [390649, 390564, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 390651, F5, 13) (dual of [390651, 390566, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(585, 390651, F5, 13) (dual of [390651, 390566, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(585, 390649, F5, 13) (dual of [390649, 390564, 14]-code), using
(72, 72+13, 267012)-Net over F5 — Digital
Digital (72, 85, 267012)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(585, 267012, F5, 13) (dual of [267012, 266927, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 390651, F5, 13) (dual of [390651, 390566, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(585, 390651, F5, 13) (dual of [390651, 390566, 14]-code), using
(72, 72+13, large)-Net in Base 5 — Upper bound on s
There is no (72, 85, large)-net in base 5, because
- 11 times m-reduction [i] would yield (72, 74, large)-net in base 5, but