Best Known (82, 82+28, s)-Nets in Base 9
(82, 82+28, 937)-Net over F9 — Constructive and digital
Digital (82, 110, 937)-net over F9, using
- net defined by OOA [i] based on linear OOA(9110, 937, F9, 28, 28) (dual of [(937, 28), 26126, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(9110, 13118, F9, 28) (dual of [13118, 13008, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 13122, F9, 28) (dual of [13122, 13012, 29]-code), using
- trace code [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- trace code [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 13122, F9, 28) (dual of [13122, 13012, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(9110, 13118, F9, 28) (dual of [13118, 13008, 29]-code), using
(82, 82+28, 13126)-Net over F9 — Digital
Digital (82, 110, 13126)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9110, 13126, F9, 28) (dual of [13126, 13016, 29]-code), using
- trace code [i] based on linear OA(8155, 6563, F81, 28) (dual of [6563, 6508, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 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(27) ⊂ Ce(26) [i] based on
- trace code [i] based on linear OA(8155, 6563, F81, 28) (dual of [6563, 6508, 29]-code), using
(82, 82+28, large)-Net in Base 9 — Upper bound on s
There is no (82, 110, large)-net in base 9, because
- 26 times m-reduction [i] would yield (82, 84, large)-net in base 9, but