Best Known (37−13, 37, s)-Nets in Base 81
(37−13, 37, 88573)-Net over F81 — Constructive and digital
Digital (24, 37, 88573)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 88573, F81, 13, 13) (dual of [(88573, 13), 1151412, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8137, 531439, F81, 13) (dual of [531439, 531402, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8137, 531439, F81, 13) (dual of [531439, 531402, 14]-code), using
(37−13, 37, 265722)-Net over F81 — Digital
Digital (24, 37, 265722)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8137, 265722, F81, 2, 13) (dual of [(265722, 2), 531407, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8137, 531444, F81, 13) (dual of [531444, 531407, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(8137, 531444, F81, 13) (dual of [531444, 531407, 14]-code), using
(37−13, 37, large)-Net in Base 81 — Upper bound on s
There is no (24, 37, large)-net in base 81, because
- 11 times m-reduction [i] would yield (24, 26, large)-net in base 81, but