Best Known (36, 36+13, s)-Nets in Base 27
(36, 36+13, 88574)-Net over F27 — Constructive and digital
Digital (36, 49, 88574)-net over F27, using
- net defined by OOA [i] based on linear OOA(2749, 88574, F27, 13, 13) (dual of [(88574, 13), 1151413, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2749, 531445, F27, 13) (dual of [531445, 531396, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2749, 531445, F27, 13) (dual of [531445, 531396, 14]-code), using
(36, 36+13, 332648)-Net over F27 — Digital
Digital (36, 49, 332648)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2749, 332648, F27, 13) (dual of [332648, 332599, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using
(36, 36+13, large)-Net in Base 27 — Upper bound on s
There is no (36, 49, large)-net in base 27, because
- 11 times m-reduction [i] would yield (36, 38, large)-net in base 27, but